UC-NRLF 


HA 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

» 

"Received        JAN      6    f8g3 .. ...  189. 
Accessions  No.i^Q  I  O      .  Clots  No. 


How  to  Teach  Elementary  Arithmetic 


.GRUBE'S  METHOD 


TEACHING  ARITHMETIC  EXPLAINED 


WITH  A   LARGE  NUMBER  OF 


PRACTICAL  HINTS  AND  ILLUSTRATIONS 


BY 

F.    LOUIS   SOLDAN 

PRINCIPAL   OF  THE  ST.   LOUIS   NORMAL  SCHOOL 


THE   INTERSTATE   PUBLISHING   COMPANY 
BOSTON:  30  FRANKLIN   STREET 


COPYRIGHT,  1878, 
Bv  VAILE  &   W1NCHELL. 


PREFACE. 


THE  first  of  the  following  two  essays  is  the  same  in  sub- 
stance as  the  one  read  before  the  St.  Louis  Teachers'  Associa- 
tion in  1870,  which  has  been  republished  since  extensively  in 
state  and  city  school  reports  and  educational  magazines.  It 
is  presented  here  in  a  somewhat  changed  form,  because  the 
practical  experience  in  the  schoolroom  has  shown  since  what 
points  of  the  method  are  in  such  harmony  with  established 
views  as  to  require  no  further  explanation,  and  what  details 
need  full  comment  and  amplification  in  order  to  guard  against 
such  mistakes  as  are  likely  to  creep  in.  In  some  respects  I 
was  guided  by  many  inquiries  on  the  part  of  the  friends  of  the 
method.  I  regret  to  say  that  I  have  not  always  been  able  to 
answer  these  questions  as  fully  as  I  wished.  I  hope  that  my 
correspondents  will  find  the  desired  explanation  in  this  new 
version  of  the  old  essay.  I  deem  it  my  duty,  however,  to  say, 
in  justice  to  Mr.  Grube,  that  the  following  pages  are  not  in 
every  respect  a  translation  from  his  work,  as  has  been  supposed 
by  some.  One  gentleman  has  done  me  the  credit  to  publish 
my  essay  over  his  own  signature  as  a  translation  from  Mr. 
Grube's  work.  It  should  be  distinctly  understood  that  the  full 
credit  for  one  and  every  idea  contained  herein  belongs  to 

3 


4  PREFACE. 

Mr.  Grube,  but  that  he  is  not  responsible  at  all  for  the  many 
imperfections  in  the  manner  in  which  his  thoughts  are  stated 
here.  In  a  few  instances  only,  the  writer  has  allowed  himself 
to  depart  from  Mr.  Grube 's  ideas.  The  two  essays  are,  may 
I  be  allowed  to  repeat,  not  altogether  a  translation,  but  rather 
an  attempt  to  give  a  condensed  account  of  the  160  pages  of 
Mr.  Grube's  work. 

The  second  essay  was  read  before  the  St.  Louis  Normal 
School  Association  in  1876,  when  it  appeared  proper  to  supply 
the  continuation  of  the  course  recommended  by  a  method 
which  had  attracted  the  attention  of  many  thinking  educators 
of  the  land,  from  California  (See  San  Francisco  Report  of 
1872)  to  New  Hampshire  (See  State  Report  of  1876).  The 
second  essay  contains  a  recapitulation  and  continuation  of  the 
first  essay.  It  presumes  as  little  as  its  predecessor  to  recom- 
mend, but  simply  submits  a  new  and  important  method  to  the 
thoughtful  consideration  of  those  who  are  interested  in  the 
matter.  If  circumstances  permit,  this  little  book  will  be  fol- 
lowed by  a  text-book  of  Primary  Arithmetic,  based  on  Grube's 
Method. 

L.  S. 

ST.  Louis,  November,  1878. 


UNIVERSITY 


GRUBE'S  METHOD 

OF 

TEACHING   PRIMARY  NUMBERS. 


THE  old,  long-established  method  in  arithmetic  is 
calculated  to  teach  the  first  four  processes  of  addition, 
subtraction,  multiplication,  division,  in  the  order  in 
which  they  are  named,  finishing  addition  with  small 
and  large  numbers,  before  subtraction  is  begun,  and  so 
forth.  A  more  recent  improvement  on  this  method 
consisted  in  excluding  the  larger  numbers  altogether  at 
the  beginning,  and  dividing  the  numbers  on  which  the 
first  four  processes  were  taught,  into  classes,  or  so-called 
circles.  The  child  learns  each  of  the  four  processes 
with  the  small  numbers  of  the  first  circle  (i.e.,  from  I 
to  10)  before  larger  numbers  are  considered ;  then  the 
same  processes  are  taught  with  the  numbers  of  the 
second  circle,  from  10  to  100,  then  of  the  third,  from 
100  to  1,000,  and  so  forth. 

Grube,  however,  went  beyond  this  principle  of  classi- 
fication. He  discarded  the  use  of  large  numbers,  hun- 
dreds and  thousands,  at  the  beginning  of  the  course,  as 
others  had  done  before  him  ;  but  instead  of  dividing 
the  primary  work  in  arithmetic  into  three  or  four  circles 
or  parts  only,  i.e.,  from  I  to  10,  10  to  100,  etc.,  he  con- 
sidered each  number  as  a  circle  or  part  by  itself,  and 
taught  it  by  a  method  that  is  to  be  set  forth  in  the 

s 


6  G RUBE'S  METHOD. 

following  pages.  He  recommended  that  the  child 
should  learn  each  of  the  smaller  numbers  in  succession, 
and  all  the  operations  within  the  range  of  each  number, 
before  proceeding  to  the  next  higher  one,  addition,  sub- 
traction, multiplication,  and  division,  before  proceeding 
to  the  consideration  of  the  next  higher  number. 

In  order  to  guard  against  a  mistake  .which  has  been 
made  rather  frequently,  it  should  be  stated  that  such 
examples  only  are  considered  to  be  within  the  limit  of 
a  number,  and  are  to  be  taught  in  connection  with  it, 
in  which  a  larger  number  than  the  one  that  is  being 
considered  does  not  appear  in  any  way  whatsoever. 
Thus,  for  instance,  when  the  number  four  is  taught,  the 
teacher  should  exclude  at  the  beginning  addition  and 
subtraction  by  fours,  multiplication  with  4  as  one  of  the 
factors,  division  with  4  as  the  divisor,  because  these 
belong  to  a  later  and  more  advanced  part  of  the  course, 
since  they  involve  in  the  sum,  minuend,  product,  or 
dividend  numbers  beyond  the  limit  of  the  one  that  is 
being  considered.  But  all  the  examples  that  do  not 
involve  a  higher  number  than  four,  are  illustrated  and 
taught,  before  passing  over  to  the  next  higher  number, 
five.  Treating,  for  instance,  the  number  2,  Grube  leads 
the  child  to  perform  all  the  operations  that  are  possible 
within  the  limits  of  this  number,  i.e.,  all  those  that  do 
not  presuppose  the  knowledge  of  any  higher  number, 
no  matter  whether  in  the  usual  classification  these  oper- 
ations are  called  addition,  subtraction,  multiplication,  or 
division.  The  child  has  to  see  and  to  keep  in  mind  that 

1  +  1=2,     2X1  =  2,     2  —  1  =  1,     2-5-1  =  2,  etc. 
The  whole  circle  of   operations  up  to  2   is   exhausted 


TEACHING  ARITHMETIC  EXPLAINED.  J 

before  the  child  proceeds  to  the  consideration  of  the 
number  3,  which  is  to  be  treated  in  the  same  way. 

Why  adhere  to  the  abstract  division  of  the  work  in 
arithmetic  into  addition,  subtraction,  etc.,  in  the  primary 
grade,  where  these  distinctions  do  not  help  to  make  the 
subject  any  clearer  to  the  pupil  ?  The  first  four  pro- 
cesses are  naturally  connected,  and  will  appear  so  in 
the  untaught  mind.  If  you  take  away  i  from  2,  and  I 
remains,  the  child,  in  knowing  this,  also  understands 
implicitly  the  opposite  process  of  adding  i  to  I  and  its 
result. 

Multiplication  and  division  are,  in  the  same  way, 
nothing  but  another  way  for  adding  and  subtracting,  so 
that  we  might  say  one  operation  contains  all  the  others. 
"Every  text -book  of  primary  arithmetic  professes  to 
teach  the  numbers  in  some  way  or  other,"  says  Grube  ; 
"  but  to  know  a  number  really  means  to  know  also  its 
most  simple  relations  to  those  numbers,  at  least,  which 
are  smaller  than  it."  Any  child,  however,  who  knows 
a  number  and  its  relations,  must  be  also  able  to  perform 
the  operations  of  adding,  subtracting,  etc.,  for  they  are 
nothing  but  the  expression  of  the  relation  in  which  one 
number  stands  to  others.  Each  example  shows  what 
must  be  added  to  or  subtracted  from  a  number  to  raise 
it  or  lower  it  to  equality  with  another,  or,  as  in  multi- 
plication and  division,  it  sets  forth  the  multiple  relation 
of  two  numbers. 

The  four  processes  are  the  direct  result  of  comparing, 
or  "measuring,"  as  Grube  calls  it,  two  numbers  with 
each  other.  Only  when  the  child  can  perform  all  these 
operations,  for  instance,  within  the  limits  of  2,  can  it 
be  supposed  really  to  have  a  perfect  knowledge  of  this 
number.  So  Grube  takes  up  one  number  after  the 


8  G RUBE'S  METHOD. 

other,  and  compares  it  with  the  preceding  ones,  in 
all  imaginable  ways,  by  means  of  addition,  subtrac- 
tion, multiplication,  and  division.  This  comparing  or 
"measuring"  takes  place  always  on  external,  visible 
objects,  so  that  the  pupil  can  see  the  objects,  the  num- 
bers of  which  he  has  to  compare  with  each  other.  The 
adherents  of  this  method  claim  for  it  that  it  is  based  on 
a  sound  philosophical  theory,  and  that  it  has  proved 
superior  in  practice  to  the  methods  in  use  before  its 
invention. 

Some  of  the  most  important  principles  of  this  method 
of  instruction  are  given  by  Grube  in  the  following : 

"i  (Language).  We  cannot  impress  too  much  upon  the 
teacher's  mind,  that  each  lesson  in  arithmetic  must  be  a  lesson 
in  language  at  the  same  time.  This  requirement  is  indispen- 
sable with  oar  method.  As  the  pupil  in  the  primary  grade 
should  be  generally  held  to  answer  in  complete  sentences,  loud, 
distinctly,  and  with  clear  articulation,  so  especially  in  arithme- 
tic, the  teacher  has  to  insist  on  fluency,  smoothness,  and  neat- 
ness of  expression,  and  should  lay  special  stress  upon  the 
process  of  solution  of  each  example.  As  long  as  the  language 
for  the  number  is  not  perfect,  the  idea  of  the  number  is  defec- 
tive as  well.  An  example  is  not  finished  when  the  result  has 
been  found,  but  when  it  has  been  solved  in  a  proper  way. 
Language  is  the  only  test  by  which  the  teacher  can  ascertain 
whether  the  pupils  have  perfectly  mastered  any  step  or  not. 

"2  (Questions).  Teachers  should  avoid  asking  too  many 
questions.  Such  questions,  moreover,  as,  by  containing  half 
the  answer,  prompt  the  scholar,  should  be  omitted.  The 
scholar  must  speak  himself  as  much  as  possible. 

"3  (Class  and  Individual  Recitation).  In  order  to  animate 
the  lesson,  answers  should  be  given  alternately  by  the  scholars 
individually,  and  by  the  class  in  concert.  The  typical  numeri- 


TEACHING  ARITHMETIC  EXPLAINED.  9 

cal  diagrams  (which,  in  the  following,  will  continually  re-ap- 
pear) are  especially  fit  to  be  recited  in  concert. 

"4  (Illustrations).  Every  process  and  each  example  should 
be  illustrated  by  means  of  objects.  Fingers,  lines,  or  any  other 
objects  will  answer  the  purpose,  but  objects  of  some  kind  must 
always  be  presented  to  the  class. 

"5  (Comparing  and  Measuring).  The  operation  of  each 
new  stage  consists  in  comparing  or  measuring  each  new  num- 
ber with  the  preceding  ones.  Since  this  measuring  can  take 
place  either  in  relation  to  difference  (arithmetical  ratio),  or  in 
relation  to  quotient  (geometrical  ratio),  it  will  be  found. to 
comprise  the  first  four  rules.  A  comparison  of  two  numbers 
can  only  take  place  by  means  of  one  of  the  four  processes. 
This  comparison  of  the  two  numbers,  illustrated  by  objects, 
should  be  followed  by  exercises  in  the  rapid  solving  of  prob- 
lems and  a  view  of  the  numerical  relations  of  the  numbers 
just  treated,  in  more  difficult  combinations.  The  latter  offer  a 
good  test  as  to  whether  the  results  of  the  examination  of  the 
arithmetical  relations  of  the  number  treated  have  been  con- 
verted into  ideas  by  a  process  of  mental  assimilation.  In  con- 
nection with  this,  a  sufficient  number  of  examples  in  applied 
numbers  are  given  to  show  that  applied  numbers  hold  the  same 
relation  to  each  other  that  pure  numbers  do. 

"6  (Writing  of  Figures).  On  neatness  in  writing  the 
figures,  the  requisite  time  must  be  spent.  Since  an  invariable 
diagram  for  each  number  will  re-appear  in  all  stages  of  this 
course  of  instruction,  the  pupils  will  soon  become  able  to 
prepare  the  work  for  each  coming  number  by  writing  its 
diagram  on  their  slates/' 


10  G RUBE'S  METHOD. 

It  will  appear  from  this  that  Mr.  Grube  subjects  each 
number  to  the  following  processes : 

I.  Exercises  on  the  pure  number,  always  using  objects  for 

illustration. 

a.  Measuring  (comparing)  the  number  with  each  of  the 

preceding  ones,  commencing  with  i,  in  regard  to 
addition,  multiplication,  subtraction,  and  division, 
each  number  being  compared  by  all  these  processes 
before  the  next  number  is  taken  up  for  comparison. 
For  instance,  6  is  first  compared  with  i  by  means  of 
addition,  multiplication,  subtraction,  and  division, 

(i  +  i  +,  etc.  =  6 ;  6  X  i  =  6 ; 

6—1  —  1,  etc.  =  i ;  6  -j-  i  =  6) 

then  with  2,  then  with  3,  and  so  forth. 

b.  Practice  in  solving  the  foregoing  examples  rapidly. 

c.  Finding  and  solving   combinations  of  the   foregoing 

examples. 

II.  Exercises  on  examples  with  applied  numbers. 


In  the  following,  Mr.  Grube  gives  but  the  outline, 
the  skeleton  as  it  were,  of  his  method,  trusting  that  the 
teacher  will  supply  the  rest.  The  sign  of  division,  as 
will  be  explained  below,  should  be  read  at  the  begin- 
ning :  "  From  ...  I  can  take  away  ...  —  times."  By 
this  way  of  reading,  the  connection  between  subtraction 
and  division  becomes  evident. 


TEACHING  ARITHMETIC  E'XPLAINED.  II 

FIRST   STEP. 
THE     NUMBER     ONE. 

"As  arithmetic  consists  in  reciprocal  ' measuring'  (com- 
paring), it  cannot  commence  with  the  number  i,  as  there  is 
nothing  to  measure  it  with,  except  itself  as  the  absolute 
measure." 

I.  The  abstract  (pure)  number. 
One  finger,  one  line ;  one  is  once  one. 
The  scholars  learn  to  write  : 

•         I. 
•  1X1   =   1. 

II.  The  applied  number. 

What  is  to  be  found  once,  in  the  room,  at  home,  on  the 
human  body? 


SECOND    STEP. 
THE     NUMBER    TWO. 


I.  The  pure  number. 
a.  MEASURING  (comparing). 


2. 


1  +    1    =    2. 
2X1    =    2. 

2  —    1    =    1. 

2-5-1  =  2.    (Read  :  From  2  I  can  take' 
away  i  twice.) 

2  is  one  more  than  i. 

1  is  one  less  than  2. 

2  is  the  double  of  i,  or  twice  i. 
i  is  one-half  of  2. 


12  G RUBE'S  METHOD. 

b.  PRACTICE  IN  SOLVING  EXAMPLES  RAPIDLY,     i  +  i  =  ? 

2    —    I    =    ?        2    -r-    I    =    ?        I    +     I    —    I     X     2    =    ?    etC. 

c.  COMBINATIONS. 

What  number  is  contained  twice  in  2  ? 
2  is  the  double  of  what  number? 
Of  what  number  is  i  one-half? 
Which  number  must  I  double  to  get  2  ? 
I  know  a  number  that  has  in  it  one  more  than  one. 
Which  is  it? 

What  number  have  I  to  add  to  i  in  order  to  get  2  ? 

II.  Applied  numbers. 

Fred  had  two  dimes,  and  bought  cherries  for  one  dime. 
How  many  dimes  had  he  left? 

A  slate-pencil  costs  i  cent.  How  much  will  two  slate-pen- 
cils cost? 

Charles  had  a  marble,  and  his  sister  had  twice  as  many. 
How  many  did  she  have? 

How  many  one-cent  stamps  can  you  buy  for  2  cents  ? 


THIRD   STEP. 
THE   NUMBER    THREE. 

I.  The  pure  number. 
a.  MEASURING. 
(i)  By  i. 

•     •     •        3- 


1  +  1  +  1  =  3. 
3x1  =  3. 

3    —    I    —    1    =  1.       (Better  than  3-1  —  1  —  1=0.) 

for,  3  -  i  = 


TEACHING   ARITHMETIC  EXPLAINED.  13 

This  ought  to  be  read :  From  3  I  can  take  away  i  3  times, 
or,  in  three,  i  is  contained  three  times.  The  ideas  of  "to  be 
taken  away  "  and  "  to  be  contained  "  must  always  precede  the 
higher  and  more  difficult  conception  of  dividing. 

(2)  Measuring  by  2. 

2  +  1  =  3;  I-f-2  =  3- 
1X2  +  1  =  3. 

3  -  2  =  i  ;  3  -  i  =  2. 
3-5-2  =  1    (i  remainder). 

(From  3,  I  can  take  away  2  once,  and  i  will  remain ;  or,  In 
three,  2  is  contained  once  and  one  over.) 

3  is  i  more  than  2,  3  is  2  more  than  i. 

2  is  i  less  than  3,  2  is  i  more  than  i. 
i  is  2  less  than  3,  i  is  i  less  than  2. 

3  is  three  times  i. 

i  is  the  third  part  of  3. 

i  and  i  are  equal  numbers,  i  and  2,  as  well  as  2  and  3 
are  unequal. 

Of  what  equal  or  what  unequal  numbers  does  3  consist, 
therefore?  etc. 

b.  PRACTICE  IN  SOLVING  EXAMPLES  RAPIDLY. 

How  many  are  3  —  1  —  1  +  2  divided  by  i  ? 

i  +  i  +  i  —  2  +  I  +  I  —  2  +  i  +  i=? 

3X   i  —  2Xi  +  i  +  i  —  2  +  i  +  i  =  ?  etc. 

The  answers  must  be  given  immediately. 

No  mistakes  can  arise  as  to  the  meaning  of  these  exam- 
ples;  the  question  whether  3X1  —  2  means  (3  x  i)  —  2 
or  3  x  (i  —  2)  is  answered  jpy  the  fact  that  these  examples 
represent  oral  work,  and  that  it  is  supposed  that  the  operation 
indicated  by  the  first  two  numbers  (3X1)  is  completed 
mentally  before  the  next  number  is  given. 


14  G RUBE'S  METHOD. 

c.  COMBINATIONS. 

From  what  number  can  you  take  twice  i  and  still  keep  i  ? 
What  number  is  three  times  i  ? 

I  put  down  a  number  once,  and  again,  and  again  once, 
and  get  3  ;  what  number  did  I  put  down  3  times  ? 

II.  Applied  numbers. 

How  many  cents  must  you  have  to  buy  a  three-cent  stamp  ? 

Annie  had  to  get  a  pound  of  tea  for  2  dollars.  Her  mother 
gave  her  3  dollars.  How  much  money  must  Annie  bring  back  ? 

Charles  read  one  line  in  his  primer,  his  sister  read  2  lines 
more  than  he  did.  How  many  lines  did  she  read? 

If  one  slate-pencil  costs  one  cent,  how  much  will  3  slate- 
pencils  cost? 

Bertha  found  in  her  garden  3  violets,  and  took  them  to  her 
parents.  How  can  she  divide  them  between  father  and  mother  ? 


FOURTH   STEP. 
THE     NUMBER     FOUR. 


I.  The  pure  number. 
a.  MEASURING. 

(i)  By  i. 


•  i 

•  i 

•  i 

•  i 


I     +    I    -f-    I    +    I    =    4    (i+i  =  a,     2  +  1  =  3). 

4X1=4- 

4—1    —     1    —     1    =    1. 

4-5-1=4. 


TEACHING   ARITHMETIC  EXPLAINED.  15 

(2)  Measuring  by  2. 

•       •       2  f   2    +    2    =   4. 

2X2  =  4. 

4    —    2    =    2. 

4-7-2  =  2. 

(3)  Measuring  by  3. 

3  +  i  =  4,  i  +  3  =  4. 
1X3  +  1  =  4- 

4  -  3  =  i,  4  -  i  =  3- 

4  -j-  3  =  i  (i  remainder). 

(In  4,  3  is  contained  once  and  i  over ;  or  from  4  I  can  take 
away  3  once,  and  one  remains.) 

Name  animals  with  4  legs  and  with  2  legs. 

Wagons  and  vehicles  with  i  wheel,  2,  and  4  wheels.     Com- 
pare them. 

4  is  i  more  than  3,  2  more  than  2,  3  more  than  i. 

3  is  i  less  than  4,  i  more  than  2,  2  more  than  i. 
2  is  2  less  than  4,  i  less  than  3,  i  more  than  i. 

i  is  3  less  than  4,  2  less  than  3,  i  less  than  2. 

4  is  4  times  i,  twice  2. 

i  is  the  fourth  part  of  4,  2  one-half  of  4. 
Of  what  equal  and  unequal  numbers  can  we  form  the  num- 
ber 4  ? 

b.  PROBLEMS  FOR  RAPID  SOLUTION. 

2x2  —  3  +  2  x  i  —  i  —  2  +  2=? 

4—1  —  1  +  1  +  1  —  3,  how  many  less  than  4?  etc. 

c.  COMBINATIONS. 

What  number  must  I  double  to  get  4  ? 

Four  is  twice  what  number? 

Of  what  number  is  2  one-half? 

Of  what  number  is  i  the  fourth  part? 

\Vhat  number  can  be  taken  twice  from  4  ? 


1 6  G RUBE'S  METHOD. 

What  number  is  3  more  than  i  ? 

How  much  have  I  to  add  to  the  half  of  4  to  get  4  ? 

Half  of  4  is  how  many  times  one  less  than  3  ?  etc. 

II.  Applied  numbers. 

Caroline  had  4  pinks  in  her  flower-pot,  which  she  neglected 
very  much.  For  this  reason,  one  day  one  of  the  flowers  had 
withered,  the  second  day  another,  and  the  following  day  one 
more.  How  many  flowers  did  Caroline  keep  ? 

How  many  dollars  are  2  +  2  dollars? 

Three  apples  and  one  apple  ? 

4  quarts  =  i  gallon. 

Annie  bought  a  gallon  of  milk ;  how  many  quarts  did  she 
have  ? 

She  paid  i  dime  for  the  quart ;  how  many  dimes  did  she  pay 
for  the  gallon  ? 


•  quart, 

•  quart, 

•  quart, 


dime, 
dime, 
dime. 


quart,  •  dime. 


What  part  of  i  gallon  is  i  quart  ? 

If  i  quart  costs  2  dimes,  can  you  get  a  gallon  for  4  dimes? 
A  cook  used  a  gallon  of  milk  in  4  days.     How  much  did  she 
use  each  day? 


The  recitations  should  be  made  interesting  and  animated  by 
frequently  varying  the  mode  of  illustration,  and  in  this  the  in- 
genuity of  the  teacher  and  her  inventive  power  can  display 
themselves  to  their  best  advantage.  It  is,  of  course,  superfluous 
to  describe  the  infinite  variety  of  objects  which  may  be  used, 


TEACHING   ARITHMETIC  EXPLAINED  I'/ 

but  a  few  suggestions  will  perhaps  prove  acceptable.  Those 
illustrations  which  compel  the  whole  class  to  be  active*  or  which 
are  of  special  interest,  and  arouse  the  attention  of  pupils,  are 
of  greater  value  than  others.  For  instance  : 

"  Class,  raise  two  fingers  of  your  right  hand ;  two  fingers  of 
your  left  hand.  How  many  fingers  have  you  raised?  Two 
fingers  and  two  fingers  are  how  many?  Two  and  two  are  how 
many  ?  Carrie  may  show  to  the  class,  with  her  fingers,  that  two 
and  two  are  four." 

This  plan  of  illustrating  should  be  used  very  frequently,  as  it 
requires  the  whole  class  to  be  active.  The  following  illustration 
is  also  commendable,  as  it  hardly  ever  fails  to  enlist  the  inter- 
est of  the  class ;  every  pupil  likes  to  be  allowed  to  illustrate  a 
problem  in  this  way  : 

"  From  four  I  can  take  away  two,  how  many  times  ?  Emma 
may  show  that  her  answer  is  correct,  by  making  some  of  the 
other  girls  stand."  (The  class  know  that  those  whom  Emma 
teaches  must  stand  until  she  makes  them  take  their  seats  again.) 
Emma :  "  Four  little  girls  are  standing  here.  From  4  little 
girls  I  can  take  away  2  little  girls  once  (making  two  of  the  four 
take  their  seats),  twice  (making  the  other  two  sit  down).  From 
4  little  girls,  I  can  take  away  2  little  girls  twice.  From  4  I  can 
take  away  2  twice.  4-5-2  =  2." 


i8 


GRUBE'S  METHOD. 


FIFTH  STEP. 
THE     NUMBER     FIVE. 


I.  The  pure  number. 
a.  MEASURING. 
(i)  By  i. 


I 
I 

I 

(2)  By  2. 

•  •       2 

•  I 

(3)  By  3. 

•  •     •     3 

•  •  2 

(4)  By  4. 

•  •     •     •     4 

•  i 


1  +  1  +  1  +  1  +  1  =  5, 

5Xi  =  5. 

5  —  1  —  1  —  1  —  1  =  1, 

5  -*-  i  =  5- 


2X2 


1=5 


5-2-2=1. 

5  -r-  2  =  2  (i  remainder). 


1X3  +  2  =  5. 

5-3  =  2;5-2  =  3. 

5-^3=  i  (2  remainder). 


4  +  i  =  5;  i  +  4  =  5. 

1X4  +  1  =  5- 

5  -  4  =  i  ;  5  -  i  =  4- 
5-7-4=1(1  remainder). 


The  fingers  are  the  best  means  of  illustration  here  :  "  Hold 
up  your  left  hand.     How  many  fingers  are  you  holding  up  ? 


TEACHING  ARITHMETIC  EXPLAINED.  19 

Hold  the  thumb  away  from  the  other  fingers.  How  many 
fingers  here?  (i)  ;  here?  (4).  i  finger  taken  from  5  fingers 
leaves  how  many  fingers  ?  i  from  5  =  ?  4  fingers  -f-  i  finger 
=  ?  4  -h  i  =  ?  Hold  your  first  finger  and  the  thumb  away 
from  the  other  fingers.  5  —  2  =  ?  3  +  2  =  ?  2  +  3  =  ?" 
etc. 

5  is  one  more  than  4,  5  is  2  more  than  3,  5  is  3  more  than 
2,  5  is  4  more  than  i.  (All  the  solutions  of  these  examples 
are  the  result  of  observation  from  illustrations  placed  before 
the  eyes  of  the  class ;  without  them  this  kind  of  instruction  is 
worthless.) 

4  is  i  less  than  5  ;  4  is  i  more  than  3,  etc. 
3  is  2  less  than  5,  etc. 

5  =  5X1- 

i  =  i  X  5  (i  is  the  fifth  part  of  five). 
5  consists  of  two  unequal  numbers,  3  +  2.     5  consists 
of  two  equal  numbers  and  one  unequal  number,  2  +  2  +  1. 

b.  PRACTICE  IN  THE  RAPID  SOLUTION  OF  EXAMPLES. 

(It  would  be  a  great  mistake  to  drill  on  the  same  example 
until  the  pupils  can  remember  it.  Such  a  practice  would  be 
worse  than  valueless ;  every  example  should  be  a  new  one  to 
the  pupil,  and  the  faculty  appealed  to  should  be  judgment  as 
well  as  memory.) 

5  —  2  —  3  +  2X2,  one-half  of  it  less  i,  taken  5 
times  =  ? 

2x2  +  1  —  3X1X2  —  3  +  4=?  etc. 

c.  COMBINATIONS. 

What  number  is  one  fifth  of  5  ?  How  many  must  I  add  to 
3  to  get  5  ?  How  many  must  be  taken  away  from  5  to  get  3  ? 
How  many  times  two  have  I  added  to  i  in  order  to  get  5  ?  I 
have  taken  away  twice  2  from  a  certain  number,  and  i  remained. 
What  number  was  it  ?  etc. 


20 


G RUBE'S  METHOD. 


II.  Applied  numbers. 

How  many  gallons  are  2  quarts  ? 

Charles  had  5  dimes ;  he  bought  two  copy-books,  each  of 
which  cost  two  dimes.  What  money  did  he  keep  ?  (This  the 
teacher  must  make  plain  by  means  of  lines  and  dots.) 


i  i 

Dime.        Dime. 


Copy-book. 


i  i 

Dime.       Dime. 

Copy-book. 


i 
Dime. 


Henry  read  a  lesson  three  times,  Emma  read  it  as  many 
times  as  he  did,  and  two  times  more.  How  often  did  she  read 
it  ?  Father  had  five  peaches,  and  gave  them  to  his  3  children. 
The  youngest  one  received  one  peach ;  how  many  did  each  of 
the  other  children  receive  ?  etc. 


SIXTH    STEP. 


THE     NUMBER     SIX. 

I.  The  pure  number. 

a.  MEASURING. 
aa  with  i 
bb  with  2 
cc  with  3 
d£/with  4 
ee  with  5  j 

ff  Miscellaneous  examples. 

b.  RAPID  SOLUTION  OF  PROBLEMS. 

c.  COMBINATIONS  OF  NUMBERS. 


Each  process  illustrated  by  six  lines,  of 
which  as  many  are  placed  in  a  row  as  are 
indicated  by  the  number  by  which  6  is  to 
be  measured. 


II.  The  applied  number. 


TEACHING  ARITHMETIC  EXPLAINED.  21 

Grube  thinks  that  one  year  ought  to  be  spent  in  this 
way  on  the  numbers  from  I  to  10.  He  says,  "  In  the 
thorough  way  in  which  I  want  arithmetic  taught,  one 
year  is  not  too  long  for  this  most  important  part  of  the 
work.  In  regard  to  extent,  the  scholar  has  not,  appar- 
ently, gained  very  much  —  he  knows  only  the  numbers 
from  I  to  10.  But  he  knows  them." 

In  reference  to  the  main  principles  to  be  observed, 
he  demands,  first,  "  that  no  new  number  shall  be  com- 
menced before  the  previous  one  is  perfectly  mastered ;" 
secondly,  "that  reviews  should  frequently  and  regularly 
take  place  ;"  and  lastly,  "that  whatever  knowledge  has 
been  acquired  and  fully  mastered  by  illustration  and 
observation,  must  be  thoroughly  committed  to  memory." 
"  In  the  process  of  measuring,  pupils  must_  acquire  the 
utmost  mechanical  skill."  It  is  essential  to  this  method, 
that  in  the  measuring,  which  forms  the  basis  for  all 
subsequent  operations,  the  pupils  have  before  their  eyes 
a  diagram  illustrating  the  process.  It  matters  not  by 
means  of  what  objects  the  pupils  see  the  operation  illus- 
trated, whether  fingers,  lines,  or  dots,  but  they  certainly 
must  see  it. 

It  is  a  feature  of  this  method,  that  it  teaches  by  the 
eye  as  well  as  by  the  ear,  while  in  most  other  methods 
arithmetic  is  taught  by  the  ear  alone.  If,  for  instance, 
the  child  is  to  measure  7  by  the  number  3,  the  illustra- 
tion to  be  used  is  : 


If  lines  or  dots  are  arranged  in  this  way,  and  im- 
pressed upon  the  child's  memory  as  depicting  the  rela- 


22  G RUBE'S  METHOD. 

tion  between  the  numbers  3  and  7,  it  is,  in  fact,  all 
there  is  to  know  about  it.  Instead  of  teaching  all  the 
variety  of  possible  combinations  between  3  and  7,  it  is 
sufficient  to  make  the  child  keep  in  mind  the  above 
picture.  The  first  four  rules,  as  far  as  3  and  7  are  con- 
cerned, are  contained  in  it,  and  will  result  from  express- 
ing the  same  thing  in  different  words,  or  describing  the 
picture  in  different  ways.  Looking  at  the  picture,  the 
child  can  describe  it,  or  read  it  as  : 

3  +  3  +  1  =  7,  or  2x3  +  1=  1, 

or  7  -  3  ~  3  =  i;  7  -*-  3  =  *  (0- 

The  latter  process  to  be  read,  From  7  I  take  away  3 
twice,  and  I  remains ;  or,  7  contains  3  twice  and  one 
more. 

Let  the  number  to  be  measured  be  10,  and  the  num- 
ber by  which  it  is  to  be  measured  be  4 ;  then  since  the 
way  to  arrange  the  lines  or  dots  for  illustration  is  to 
have  as  many  dots  or  lines  as  are  indicated  by  the 
larger  number,  and  as  many  of  them  in  a  row  as  are 
indicated  by  the  smaller  number,  we  write  : 


The  child  will  be  able  to  see  at  once,  by  reading  the 
diagram,  as  it  were,  that 

4  +  4  +  2  =  10 ;  2  X  4  +  2  =  10; 

10  -4-4=  2;   10  -=-4=2  (2), 

and  to  perceive  at  a  glance  a  variety  of  other  combina- 
tions.    The  children  will,  in  the  course  of  time,  learn 


TEACHING  ARITHMETIC  EXPLAINED.  2$ 

how  to  draw  these  pictures  on  their  slates  in  the  proper 
way.  Nor  will  it  take  long  to  make  them  understand 
that  every  picture  of  this  kind  is  to  be  "read"  in  four 
ways,  first  using  the  word  and,  then  times,  then  less, 
then,  From  .  .  .  can  be  taken  away  .  .  .  times.  As 
soon  as  the  pupils  can  do  this,  they  have  mastered  the 
method,  and  can  work  independently  all  the  problems, 
within  the  given  number,  which  are  required  in  measur- 
ing. 

It  would  be  a  mistake  to  suppose  that,  in  teaching 
according  to  this  method,  memory  is  not  required  on 
the  part  of  the  child.  Memory  is  as  important  a  factor 
here  as  it  is  in  all  instruction.  This  should  be  empha- 
sized, because  with  some  teachers  it  has  become  almost 
a  crime  to  say  that  memory  holds  its  place  in  education. 
To  have  a  good  memory,  is,  in  their  eyes,  a  sign  of 
stupidity.  Grube  was  too  experienced  a  teacher  to  fall 
into  this  error.  While  by  his  method  the  results  are 
gained  in  an  easier  and  more  natural  way,  whatever 
result  is  arrived  at  must  be  firmly  retained  by  dint  of 
memory  assisted  by  frequent  reviews. 

(END  OF  FIRST  ESSAY.) 


24  G RUBE'S  METHOD. 


NUMBERS    ABOVE    TEN. 


SECOND  ESSAY. 

WHEN  Grube's  Method  of  teaching  the  elements  of 
arithmetic  was  first  introduced  to  the  Teachers'  Asso- 
ciation of  St.  Louis,  in  1870,  it  was  not  presented  with 
the  assurance  of  warranted  success  as  the  only  plan  of 
teaching  this  important  study,  but  rather  as  an  attempt 
to  demonstrate  practically,  in  a  given  instance,  to  some 
extent  at  least,  how  far  methods  of  teaching  may  be 
redeemed  from  the  bane  of  vagueness,  which,  as  long 
as  it  lasts,  excludes  them  from  the  rank,  in  the  science 
of  Pedagogics,  to  which  they  might  otherwise  be  en- 
titled. Grube's  Method  was  submitted  with  diffidence 
to  the  judgment  of  practical  teachers,  without  the  com- 
mendation of  any  champion  who  expressed  an  implicit 
belief  in  its  immediate  signal  success.  We  may  speak 
disparagingly  of  the  often  frivolous  distinction  between 
theory  and  practice,  which  ignores  the  harmonious 
parallelism  between  the  world  of  thought  and  the  world 
of  fact,  but  we"  shall  nevertheless  insist  upon  practical 
usefulness  as  the  test  of  any  psychologically  correct 
method  of  teaching. 

To-day  Grube's  Method  of  teaching  arithmetic  does 
not  lack  friends  and  supporters  :  it  has  been  tried  and 
adopted,  not  in  one  city  alone,  but  has  become  recog- 
nized throughout  the  country.  Long  before  its  practi 


TEACHING  ARITHMETIC  EXPLAINED.  25 

cal  test  in  the  district  schools  of  St.  Louis,  it  was  made 
part  of  the  regular  course  of  instruction  in  the  schools 
of  San  Francisco,  and  many  other  cities  have  adopted  it 
since.  Practical  experience  has  shown  the  advantages 
and  disadvantages  of  this  system  as  far  as  the  part 
which  was  presented  at  that  time  is  concerned ;  namely, 
the  numbers  from  one  to  ten  only. 

Beyond  this  limit  there  is  still  disputed  ground,  and 
it  may  be  allowable  to  say  that  the  continuation  is  of- 
fered to-day  in  the  same  spirit  as  the  beginning  was 
years  ago.  It  is  simply  a  report  on  an  ingenious  meth- 
od which  is  considered  worthy  the  notice  of  thoughtful 
teachers,  and  which  seems  to  deserve  a  fair  trial,  con- 
tinued for  a  sufficient  length  of  time  to  extend  beyond 
the  period  during  which  a  new  method  seems  objection- 
able because  it  is  new,  and  hence  collides  with  a  prac- 
tice which  habit  has  made  convenient. 

The  leading  idea  is  the  same  throughout  Grube's 
Method.  To  show  the  principle  of  teaching  the  higher 
numbers  to  100  is  to  recapitulate  the  principles  that 
are  to  guide  the  teacher  in  his  treatment  of  the  num- 
bers from  i  to  10.  That  the  four  processes  are  taught 
with  each  number,  before  -the  following  one  is  consid- 
ered, forms,  no  doubt,  a  characteristic  feature  of  Grube's 
Method,  but  it  is  a  common  mistake  to  suppose  that  it 
is  the  leading  idea.  It  certainly  emanates  from  this 
idea,  but  it  is  not  the  idea  itself.  The  leading  principle 
is  rather  that  of  objective  illustration. 

In  a  very  general  way  it  may  be  said  that  in  examples 
in  primary  arithmetic  two  numbers  are  given,  and  their 
relation,  expressed  by  a  third  number,  is  to  be  found. 
Hence  the  elementary  processes  may  be  considered  as 
the  comparing  of  one  number  with  the  other,  or  the 


26  G RUBE'S  METHOD. 

measuring  of  one  by  the  other.  On  the  basis  of  this 
general  theory,  Grube  suggests  a  general  plan  of  illus- 
tration, according  to  which  the  larger  number  of  the 
two  numbers  given  is  represented  by  the  total  number 
of  lines  or  dots  placed  on  the  blackboard.  These  lines 
are  arranged  into  sets  or  groups,  each  containing  as 
many  lines  or  dots  as  are  indicated  by  the  smaller 
number  of  the  two.  Thus,  if  the  numbers  6  and  2  are 
to  be  compared  with  each  other,  the  illustration  consists 
of  six  dots,  arranged  two  by  two. 


The  measuring  of  9  by  4  is  illustrated  by  four  dots  and 
four  dots  and  one  dot. 


This  contains  the  main  principle  of  Grube's  Method. 
If  perception  has  seized  this  illustration,  and  wrought  it 
into  a  mental  picture,  the  solution  of  all  the  existing 
elementary  relations  between  the  two  numbers  has 
been  grasped  implicitly.  For  the  four  processes  are 
simply  different  interpretations  of  this  symbolic  dia- 
gram. When  this  picture  appears  before  the  mind,  it 
may  be  interpreted  as  addition  or  multiplication,  i.e., 
our  illustration  may  be  read, 

4  +  4  +  i  =  9,  or  (2  X  4)  +  i  =  9  ; 

and  by  the  retrograde  process,  when  the  illustration  is 
made  to  disappear  from  the  blackboard,  it  may  be  in- 


TEACHING  ARITHMETIC  EXPLAINED.  2/ 

terpreted  by  subtraction  or  division,  as  9  —  4  —  4  =  I, 
or  from  9  I  can  take  away  4  twice  and  leave  i.  But 
the  main  point  in  this  is,  that  the  whole  process  is  based 
on  well  selected  and  arranged  illustrations,  and  is  an 
object  lesson  on  numbers.  A  plan  of  teaching  which 
ignores  this  main  point,  and  flatters  itself  to  have  found 
the  gist  of  the  new  idea  by  jumbling  together  addition, 
subtraction,  multiplication,  and  division,  without  the 
most  extensive  use  of  illustrative  objects,  and  without 
systematic  arrangement,  has  nothing  in  common  with 
Grube's  Method.  In  the  latter,  the  clearest  order  and 
regularity  prevail  throughout.  Below  10,  each  number 
is  compared  with  the  number  i,  by  means  of  addition, 
subtraction,  multiplication,  and  division,  then  with  the 
number  2,  then  with  3,  etc.  The  pupil  will  soon  learn 
to  perceive  the  regularity  of  this  process ;  and  at  the 
moment  he  has  understood  that  part,  he  can  by  inde- 
pendent work  discover  the  primary  arithmetical  rela- 
tions of  a  number,  and  prepare  a  synopsis  or  diagram 
of  the  same. 

A  frequent  and  very  dangerous  mistake  is  the  omis- 
sion, or  neglect,  of  applied  examples.  The  pure  number 
as  the  universal  expression  of  arithmetical  truth  is  of 
the  greatest  importance,  but  the  pupil  throughout  his 
school-course  finds  the  greatest  difficulty  in  working 
with  applied  numbers.  Moreover,  arithmetic  is  studied 
for  life ;  and  in  life,  there  are  none  but  applied  examples. 
Hence,  after  the  universal,  the  pure  number,  has  been 
mastered  by  means  of  observation,  particular  applica- 
tion should  follow  immediately,  and  copious  examples, 
clothed  in  the  most  varied  forms,  should  be  solved.  The 
training  which  the  pupil  receives  from  practice  with 
applied  problems  is  different  in  kind  from  that  with  pure 


28  G RUBE'S  METHOD. 

numbers,  and  hence  cannot  be  slighted  in  the  primary 
grades  without  retarding  the  progress  in  the  higher 
classes.  Without  sufficient  practice  in  this  direction, 
there  is  danger  of  mechanical  and  dull  work,  and  the 
best  opportunities  for  the  pupil's  display  of  inventive 
ingenuity  are  lost. 

The  difficulty  which  the  study  of  arithmetic  presents 
in  the  higher  grades  lies  not  in  the  mechanical  handling 
of  numbers,  —  in  most  cases  the  pupils  succeed  very  well 
in  that,  —  but  it  lies  in  the  fact  that  the  words  of  the 
problem  puzzle  them.  The  qualitative  element  dis- 
turbs and  conceals  the  quantitative.  If  this  assertion 
is  correct,  a  great  deal  of  training  with  applied  numbers 
should  be  given  at  a  time  of  the  course  when  the  pure 
number  which  is  considered  is  so  small  as  to  allow  the 
scholar,  after  having  mastered  it,  to  concentrate  his 
whole  attention  on  the  puzzle  that  lies  in  the  wording, 
in  the  qualitative.  Wherever  sufficient  training  of  this 
kind  has  sharpened  the  wit  of  the  pupils  in  the  lower 
grades,  they  will  no  longer  consult  the  heading  of  the 
chapter  as  the  first  step  in  the  solution  of  a  problem, 
in  order  to  find  whether  it  means  addition  or  division, 
interest  or  long  measure,  and  find  themselves  in  a  help- 
less and  forlorn  condition  when  they  meet  an  example 
which  is  not  labeled  by  any  heading. 

An  analysis  of  the  operation  with  each  nifmber  shows 
as  the  two  principal  elements  : 

I.  The  number  considered  in  its  universal  quantitative 
character,  or  pure  number.     The  process  is  from  objec- 
tivity to  abstraction. 

II.  TJie  quantitative  in   special  qualitative  form,  or 
applied  number.     Here  we  proceed  from  abstraction  to 
application. 


TEACHING  ARITHMETIC  EXPLAINED.  29 

Under  the  first  or  pure  number  we  have  the  sub- 
topics : 

a.  Comparing   with,  or   measuring   by,  each    of   the 
preceding  numbers,  from  I  to   10,  considering  addition, 
multiplication,  subtraction,  and  division. 

b.  Combinations  of  the  two  numbers  treated  of,  the 
results  to  be  within  the  limits  of  the  greater  one  of 
the  two  numbers.     This  is  a  very  important  process,  no 
doubt,  but  the  temptation  lies  near  to  give  too  much 
prominence  to  it  by  forgetting  that  it  is  a  part  only  of 
Grube's  Method.     The  systematic  comparison  of  num- 
bers is  of   greater  importance,  and  it   is   an   error  to 
spend  as  much  time  on  these  combinations  as  if   the 
method  consisted  of  nothing  but  these. 

c.  Sufficient   practice   in   the   rapid    solution    of   ex- 
amples. 

In  the  former  essay,  the  treatment  of  the  numbers 
from  one  to  five  was  explained.  As  the  last  step  within 
the  circle  of  numbers  from  one  to  ten,  and  as  the 
transition  to  the  province  of  larger  numbers,  the  treat- 
ment of  the  number  ten  is  of  great  importance.  Grube 
describes  it  in  the  following  way  : 


30  G RUBE'S  METHOD. 

TENTH   STEP. 
THE     NUMBER     TEN. 

We  have  arrived  at  a  number  which  is  again  treated  as  a 
unit.  Hence  we  write  it  by  means  of  the  figure  one ;  but  to 
show  there  is  ten  times  as  much  in  this  as  in  the  figure  one 
which  we  had  before,  we  move  it  one  place  toward  the  left,  by 
which  we  mean  to  say,  This  unit  means  a  ten.  The  empty 
place  of  the  simple  unit  is  filled  out  by  a  cipher. 

I.  The  pure  number. 
a.  MEASURING  (10;  i). 


I+T+I+I+I  etc.  =  10 


10  x  i  =  10 


10  —  i  —  i  etc.  =  i 


^-  i  =  10 


(10  ;    a) 


2  +  2  etc.  =  10 
5  +  2  =  10 

IO  — 2  —  2CtC.=  2 
10-^2  =  5 


(10;  3) 

3  +  3  +  3  +  i  =  io 
3  X  3  +  i  =  10 
10-3-3-3  =  1 
10-5-3  =  3  (i) 


(10 ;  6) 


6  +  4=10 
1X6  +4  =  10 
10  —  6  =  4 
10-5-  6  =  i  (4) 


TEACHING  ARITHMETIC  EXPLAINED.  31 

Miscellaneous  Measuring : 

10  consists  of  two  equal  numbers,  5+5. 

10  consists  of  five  equal  numbers,  2  +  2  +  2+  2  +  2. 

10  consists  of  two  equal  numbers  and  one  unequal, 
5X3  +  1. 

10  consists  of  four  unequal  numbers,  1  +  2+3  +  4. 
Review  of  the  multiple  relations  within  the  number  ten. 

A.  I.  i  is  one-half  of  2,  one-third  of  3,  one-fourth  of  4,  etc. 
II.  2  is  one-half  of  4,  one-third  of  6,  etc. 

III.  3  is  one-half  of  6,  one-third  of  9. 

IV.  4  is  one-half  of  8. 
V.  5  is  one-half  of  10. 

B.  I.   10  is  10  times  i,  5  times  2,  2  times  5. 
II.  9  is  9  times  i,  3  times  3. 

III.  8  is  8  times  i,  4  times  2,  2  times  4. 

IV.  7  is  7  times  i. 

V.  6  is  6  times  i,  3  times  2,  2  times  3. 
VI.  5  is  5  times  i. 

VII.  4  is  4  times  i,  2  times  2. 

VIII.  3  is  3  times  i. 

IX.  2  is  2  times  i. 

X.  i  is  once       i. 

What  numbers  are  contained  without  any  remainder  in  10, 
9,8? 

What  numbers  have  no  other  numbers  contained  in  them 
without  remainder  except  the  number  i  ?  (The  prime  numbers 
'»  3>  5.  7-) 

b.  COMBINATIONS  (Oral  work). 

One  nickel  and  two  cents  and  three  cents,  less  6  cents,  of 
this  take  one-half  three  times,  and  add  twice  two  cents ;  how 
many  cents? 


32  G 'RUBE'S  METHOD. 

(There  is  no  better  exercise  for  rapidity  and  exactness  than 
this.  Short  combinations,  slowly  pronounced  at  first,  until  the 
class  can  solve  more  difficult  problems,  given  out  quickly.  The 
teacher  should  take  care  not  to  discourage  the  class  by  ex- 
amples that  can  be  answered  by  the  brightest  scholars  only. 
No  guessing  should  be  allowed  ;  use  illustrations.) 

(2  x  2)  -f-  (2  x  3)  -  (3  X  3)  4-  (2  X  4)  +  i  =? 
10—2  —  1  —  2  —  1  —  2  —  1=? 
1  +  2  +  3  +  4=?  etc. 

c.  PRACTICE  IN  THE  RAPID  SOLUTION  OF  EXAMPLES. 

What  number  is  i  more  than  twice  3  ? 

Twice  five  is  how  many  more  than  three  times  three  ?  than 
twice  four? 

A  father  distributed  10  apples  among  his  children,  so  that 
each  older  child  received  one  more  than  the  one  next  below 
him  in  age.  How  many  apples  did  each  child  receive?  \The 
pupils  know  that  10  consists  of  4  unequal  numbers,  i,  2,  3,  4, 
of  which  each  following  number  is  greater  by  one  than  the 
preceding.  Hence  the  father  could  divide  the  apples  so  that 
the  youngest  received  one,  the  next  two,  etc.) 

Charles  had  learned  four  words  in  spelling.  His  brother 
said,  "  I  know  twice  as  many  as  you,  and  2  more."  How  many 
did  he  know?  Solution  :  If  Charles  had  learned  4  words,  and 
his  brother  knew  twice  as  many  and  2  more,  he  knew  2X4  +  2 
words  =  10  words. 

William  said,  "  I  am  5  times  as  old  as  my  little  sister."  She 
was  2  years  old.  How  old  was  William  ?  Solution :  If  the 
little  sister  was  2  years  old,  and  William  five  times  as  old,  he 
was  5  x  2,  or  10  years  old. 

II.  Applied  numbers. 

10  days  are  how  many  weeks  and  days? 
10  cents  are  how  many  dimes?  nickels? 


TEACHING   ARITHMETIC  EXPLAINED.  33 

Fred  had  one  dime  :  he  bought  2  slate-pencils  for  one  cent 
each,  and  a  piece  of  candy  for  5  cents.  How  much  money  did 
he  spend  ?  How  much  had  he  left  ? 

One  lead-pencil  costs  5  cents;  how  much  will  two  lead- 
pencils  cost? 

How  many  marbles  at  2  cents  apiece  can  you  buy  for  10 
cents?  Solution  :  For  2  cents  I  get  i  marble,  hence  for  10  cents 
I  get  five  marbles,  since  10  cents  are  5X2  cents.  Or,  If  I 
give  to  the  store-keeper  2  cents,  he  gives  me  i  marble ;  but 
if  I  have  10  cents,  I  can  give  him  5  times  2  cents,  and  so  he 
gives  me  5  times  one  marble,  etc. 


With  this  number,  says  Grube,  the  first  and  most 
important  step  in  arithmetic  has  been  completed.  If 
the  subject  has  been  taught  as  it  should  have  been,  one 
year  is  not  too  long  a  time  for  it.  (It  seems  that  Grube 
is  speaking  on  the  supposition  that  about  two  hours  a 
week  form  the  time  given  to  the  study  of  arithmetic.) 
The  pupil's  knowledge  is  not  very  extensive  :  he  knows 
but  the  numbers  from  I  to  10.  But  would  he  really 
possess  any  knowledge  of  arithmetic  if  he  were  able  to 
count  up  to  100  and  beyond  without  being  able  to  solve 
any  problem,  even  with  the  smallest  number  ?  Learn- 
ing the  names  of  numbers  up  to  100  is  not  the  same 
as  learning  to  count,  and  is  simply  learning  by  rote  a 
series  of  words,  not  of  much  more  importance  for  arith- 
metic than  the  committing  to  memory  of  a  few  lines  of 
poetry.  Special  attention  should  be  given  to  the  prac- 
tice in  the  rapid  solution  of  miscellaneous  examples 
(I.  c.),  as  these  exercises  are  essential  for  a  clear  idea  of 
number.  If  clear  perception  has  preceded  them,  they 
will  present  no  difficulty.  Remember  that  no  new 
number  is  to  be  taken  up  before  the  previous  one  has 


34  G RUBE'S  METHOD. 

been  thoroughly  mastered,  and  that  frequent  reviews 
must  help  the  pupil  in  fixing  in  his  memory  the  princi- 
pal examples  which  have  been  considered  and  reduced 
to  writing. 

Passing  over  to  the  higher  numbers,  Grube  says,  "  In 
the  second  year  the  numbers  from  10  to  100  are  to  be 
studied.  The  following  principles  must  be  observed  in 
this  work  : " 

1.  Fingers  and  lines  are  used  for  illustration,   the  former 
being  the  most  natural  means. 

2.  The  process  with  the  numbers  from  10  to  100  is  the  same 
as  that  for  the  smaller  numbers.     Multiplication  and  division 
form  the  subjects  of  written  and  oral  work,  while  addition  and 
subtraction,  as  a  rule,  need  oral  treatment  only.     Measuring 
each  new  number  by  the  numbers  from  i  to  10  is  continued  as 
oral,  preparatory  work  until  the  pupils  have  acquired  in  it  the 
greatest  mechanical  skill. 

3.  Greatest  diversity  of  expression  and  sufficient  variety  are 
aimed  at  in  the  selection  of  examples,  in  pure  as  well  as  in 
applied  numbers,  so  that  the  pupil  may  free  himself  gradually 
from  the  uniformity  of  the  elementary  diagram  and  schedule.. 
Applied  examples  should  not  go  beyond  the  limit  of  qualitative 
relations  taken  from  daily  life  with  which  the  pupil  is  familiar. 
This  will  give  him  an  opportunity  of  inventing  examples  him- 
self, and  the  permission  to  give  an  example  to  the  class  may  be 
made  a  reward  for  that  pupil  who  succeeds  in  finding  the  solu- 
tion of  some  examples  first. 

Before  proceeding  to  describe  Grube's  treatment  of 
some  numbers  of  the  circle  from  10  to  100,  it  will  be 
best  to  recall  to  memory  the  few  essential  points  of 


TEACHING  ARITHMETIC  EXPLAINED.  35 

difference  and  agreement  with  the  previous  part  of  the 
course. 

1.  The   processes   with    each    number   remain    the    same, 
namely : 

1.  Exercises  with  the  pure  number,  by 

(a)  Comparison. 

(/£)   Combination. 

(c)   Practice  in  the  rapid  solution  of  examples. 

2.  Exercises  with  applied  number. 

2.  Objective  illustrations  form  the  most  important  part  of 
each  exercise.     Arithmetic  is  a  series   of  object   lessons   on 
numbers. 

3.  Each  new  number  is  not  compared  with  all  the  numbers 
below  itself,  but  with  the  numbers  from  i  to  10  only. 

4.  Comparison  with  these  numbers  by  means   of  addition 
and  subtraction  forms  as  a  rule  the  subject  of  oral  work  only : 
comparison  by  multiplication  and  division  is  practised   both 
orally  and  in  writing. 

5.  In  writing  out  these  comparisons  of  numbers,  the  ex- 
amples are  no  longer  placed  side  by  side,  but  below  each  other : 

(n  ;  2)  2  +  2=4 
4  +  2  =  6 
6  +  2=8 
8  +  2  =  10 

IO   +    I    =    II 

6.  Oral  comparison  by  addition  and  subtraction  takes  usually 
the    form   of,   Count   upward   or   downward   by   twos,  threes, 
fours,  etc. 


36  G RUBE'S  METHOD. 

7.  As  the  same  examples  occur  frequently,  Grube  supposes 
that  the  pupil  has  acquired  sufficient  skill  to  master  about  two 
numbers  each  recitation ;  he  is  speaking,  however,  of  recitations 
of  60  minutes  each. 

8.  More  time  is  to  be  given  to  the  lower  numbers  from  i  to 

24,  and  especially  to  numbers  that  are  of  importance  in  applied 
examples  as  representing  some  division  in  compound  numbers, 
such  as  12    (dozen,  number  of  months,  etc.),   14   (days  in  2 
weeks),  15,  1 6  (number  of  ounces  in  a  pound),  18,  20,  21,  24, 

25,  28,  30,  36,  48,  56,  64,   72,  etc.     In  connection  with  them 
the  principal  divisions  of  compound  numbers  should  be  taught. 

After  this  general  explanation,  an  application  of  the 
principles  set  forth  to  a  few  particular  numbers  will 
suffice  to  show  the  process. 


TEACHING  ARITHMETIC  EXPLAINED. 

TWELFTH   STEP. 
THE     NUMBER     TWELVE. 

I.  a.  —  Pure  number. 
MEASURING. 


10 


12 


Oral  work.  —  Measuring. 


(12;  i)           (12;  2)                    (12;  3) 

(12;  6) 

i  +  i  = 

2    +    I    = 
II    +    I    = 

'  2    +    2    = 

4  +  2  = 

6   +   2    = 

etc. 

3  +  3  = 
6  +  3  = 
9  +  3  = 

6  +  6  =  12 

or 
i,  2,  3,  4, 
etc. 

or 
2,4,6,8,  10,  12 

. 

or 
3>  6,  9,  12 

f   12    —    I    = 

II    —    I    = 
10   —    I    = 

12    —    2    = 
10   —    2    = 

12-3  = 
9-3  = 

12   -   6   = 

9    «c.= 

or 

12,    II,    10, 

etc. 

or  12,  10,  8,  6 

or 
12,  9,  6,  3 

12X1    =    12  6X2=    12  4X3=  2X6    = 

12    -5-    I    =    12          12    -5-    2    =  12    -=-   3    =         1 2    -5-   6    = 


38  G RUBE'S  METHOD. 

WRITTEN  WORK. 

12  =  12  x     i  12  =  12  x  ?  12  =  11  +  1 

=     6x2  i2=6x?  (i2isi  more 

=     4X3  i2=4X?  than  n.) 

=     3X4  i2=3X?  =10+2 

=       2X5    +    2  l)    12    =    12    X 

=     2x6  2)i2  =  6x  etc.  =9  +  3 

=     *   X     7  +  5  To  be  read  : 

=     1x8  +  4  a.  i  can  be  taken  away  from  12, 

=     1X9  +  3  I2  times. 

=     i  X  10  +  2  b.  i  is  contained  or  is  in  12,  12 

i  tenth  2  units  times. 

c.  i  is  the  1 2th  part  of  12,  etc. 

Of  what  equal  numbers  is  twelve  composed  ? 
Of  what  unequal  numbers  ? 

Give  three  numbers  that  make  twelve,  of  which  each  follow- 
ing number  is  two  more  than  the  previous  one. 

b.  —  COMBINATIONS.     (  Oral. ) 

(2  X  2)  +  (2  x  2)  +  (2  +  2)  =? 

2  +  3  +  3  +  2  +  2-4  +  4-  (4x2)=? 

Charles,  Fred,  and  George  had  1 2  apples ;  they  ate  one-half 
of  them  and  one  more  ;  how  many  had  they  left  ?  how  many  did 
they  eat?  etc. 

c.  —  PRACTICE  IN  THE  RAPID  SOLUTION  OF  EXAMPLES. 
The  third  part  of  1 2  is  what  part  of  8  ? 

One-half  of  1 2  is  how  many  times  3  ? 

What  is  the  difference  between  one-half  of  1 2  and  one-half 
of  10? 

12  is  three  times  what  number? 

What  number  must  I  take  from  1 2  to  have  9  ? 

What  number  taken  away  from  1 2  leaves  4  ?  etc. 


TEACHING  ARITHMETIC  EXPLAINED,  39 

« 

II.  Applied  number. 

12  pieces  equal  a  dozen.     Half  a  dozen  =  ? 

12  months  are  called  a  year.  (The  names  of  the  months 
are  to  be  committed  to  memory.) 

What  part  of  a  dozen  are  six  pieces  ? 

What  part  of  a  year  are  six  months  ? 

3  months  are  a  quarter  (of  a  year). 

3  pieces  are  a  quarter  of  a  dozen. 

A  month  has  about  4  weeks.  Fred  pays  $12  a  month  for 
piano  lessons  ;  how  much  does  he  pay  a  week  ? 

Solution  :  One  month  has  4  weeks.  If  he  pays  for  4  weeks 
$12,  he  pays  for  one  week  the  fourth  part  of  12,  which  is  £3. 

A  father  paid  $2  a  month  for  private  lessons  given  to  his  son. 
How  much  did  he  pay  in  a  quarter  ?  in  half  a  year  ? 

How  many  slate-pencils  at  three  cents  apiece  can  you  buy 
for  12  cents? 

Illustrate : 

•  •     •     •  000  n 

•  •      •      •  ©00  n 

•  •••  000  n 

•  •••  000  n 

Caroline  learned  by  heart  1 2  definitions  in  three  days,  etc. 
How  many  each  day?  etc. 

The  teacher  should  prepare  collections  of  such  examples  in 
writing. 

The  numbers  from  10  to  100  are  treated  in  a  similar  way; 
as  a  further  illustration,  the  treatment  of  the  number  30  is  given 
in  full.  Such  numbers  as  17,  19,  22,  23,  26,  etc.,  which  are  of 
less  importance  than  numbers  that  represent  some  frequently 
occurring  division  in  the  denomination  of  number  (12,  18,  24, 
36  =  dozen,  months,  7  =  days,  10,  15  =  cents,  etc.),  are 
treated  in  their  relation  as  pure  numbers  only,  and  the  pro- 
cesses taken  up  under  II.  are  omitted  with  them. 


40  G RUBE'S  METHOD. 

THIRTIETH    STEP. 
THE     NUMBER     THIRTY. 


(3  times  the  fingers  of  two  hands.) 

I.  a.  —  i .  CONNECTION  WITH  FORMER  STEPS  :  If  I  add  one 
unit  to  29  we  have  3  tens. 
Three  tens  are  called  thirty. 

a.  —  2.  MEASURING  BY  THE  NUMBERS  FROM  i  TO  TEN. 
Oral. 

(3<>;  i)  (30;  2) 

Count  from  i  to  30.  2,  4,  6,  8,  10,  etc. 

Count  from  30  to  i.  30,  28,  26,  24,  etc. 

(3°;  3)  (30;  4)  (30;  s) 

(30;  6)  (30;  10) 

6,  12,  18,  24,  30  10  -f  10  =  20 

20  +  10  =  30 

30,  24,  1 8,  12,  6  30  —  10  =  20 

20  —  10  =  10 

30x1=      15x2=        5x6=       3X10  =  30 

30   -f-    I    =  30    -r-    2    =  30    -r-    6    =  30    -r-    10    =       3 

30    =    29    +    I 

28  +  2  In  counting  by  2's,  3*5,  etc.,  a  pupil  should 

etc.       point  to  the  illustration.     The  teacher  should 

stop  frequently  in  this  exercise,  and  make  the 

pupils  state  how  many  tenths  and  units  they  have  counted  so 


TEACHING  ARITHMETIC  EXPLAINED.  41 

far,  and  how  many  they  have  still  to  count  up  to  30.  For  in- 
stance :  Class,  i,  2,  3,  4,  5,  6,  7,  —  "Stop."  Pupil:  "We  are 
within  the  first  ten,  three  more  are  necessary  to  complete  the 
first  ten,  23  units  to  make  up  30."  The  same  should  be  prac- 
tised in  counting  downward. 

Written  Work. 


3  X  10  —  30 

30  =  30  X 

i     30  -;-  i  =  30* 

i  =  •&  X  30  ( 

=  15  x 

2      30  -f-  2   =  15 

2  =  T5  X  30 

=  10  X 

3      -5-3  =  10 

=  7  X 

4  +  2   -=-4  =7  (2 

) 

=  6  X 

5      +5  =  6 

, 

=  5  x 

6      -6  =  5 

=  4  X 

7  +  2   -=-7  =  4(2 

) 

=    3  X  10 


10=    3 


30  is  composed  of  what  equal  numbers? 

30  is  composed  of  which  2,  3,  4,  etc.,  unequal  numbers? 

In  these  operations  the  30  dots  on  the  board  should  be  sep- 
arated into  groups  of  2,  3,  etc.,  by  placing  lines  between  them, 
i-e-  (30;  3) 


If  these  suggestions  meet  with  that  support  on  the  part  of 
teachers  which,  as  a  rule,  is  most  generously  given  to  new 

*  These  examples  are  to  be  read  by  the  pupil  in  several  ways:  a.  From  ...  I  can 
take  away  .  .  .  times,  b.  In  30  .  .  .  is  contained  .  .  .  times,  c.  The  .  .  .  th  part  of 
30  is  ... 


42  G RUBE'S  METHOD. 

methods  of  value,  it  would  be  a  good  plan  to  have  10  lines  or 
dots  painted  in  a  convenient  place  on  the  board  in  the  rooms 
of  the  lowest  grades,  and  100  lines  or  dots  arranged  10  by  10 
painted  on  the  board  in  the  next  higher  rooms.  The  chalk- 
dots  by  which  the  pupils  divide  the  lines  or  dots  into  groups 
might  then  be  wiped  off,  when  a  new  relation  is  taken  up,  with- 
out erasing  the  painted  lines. 

b.  —  COMBINATIONS  WITHIN  THE  LIMITS  OF  30. 
i  dozen  -f-  4  pieces  -f-  2  pieces  +  \  dozen  =  ? 

i  dime  -f  5  cents  +  i  dime  =  how  many  cents  ? 
(3  X  5)  +  (2  x  4)  4-  7  -  15  -  8  +  5  +  9  =  ? 
4x6,  one-half,  again  one-half,  5  times  =  ?  etc. 

c.  —  EXERCISES  IN  RAPID  CALCULATION. 

Take  19  from  30  (19  =  10+9;  30  —  10  =  20; 
20  —  9  =  ii  ;  30  —  19  =  n). 

Twice  15  (15  =  i  ten  and  5  units,  2X1  ten  =  etc.). 

Compare  30  with  16  (30  =  3  tens;  16  =  i  ten  and  6 
units ;  4  units  must  be  added  to  the  six  to  complete  the  second 
10,  and  another  ten  to  make  it  3  tens.  Hence  30  is  i  ten  and 
4  units,  or  14  more  than  16). 

II.  Applied  examples. 

(30  pieces  =  2\  dozen;  30  months  =  2\  years,  etc.) 
A  great  variety  of  these  examples  should  be  given ;  but  even 
more  important  than  this  is  the  thoroughness  with  which  each 
example  is  illustrated  and  worked  through.  Let  the  teacher 
move  quietly  in  the  stereotyped  form  of  this  method,  so  that 
the  pupil  becomes  strong  and  self-active  in  the  application 
of  the  familiar  process.  This  apparently  mechanical  form  rests 
on  self-activity,  and  leads  to  self-reliance,  self-confidence,  and 
skill.  More  pupils  fail  in  arithmetic  from  diffidence  than  from 
any  other  cause.  In  conclusion  of  the  numbers  of  the  second 
circle,  the  method  of  teaching  the  number  100  is  given. 


TEACHING  ARITHMETIC  EXPLAINED. 


43 


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_|st 

HUNDREDTH    STEP. 
NUMBER     ONE     HUNDRED. 

Considerable   time    should 
be  spent  on  the  number  100. 
Besides   the   regular  process 
which  has  been  explained  in 
^connection  with  the  number 
30,   a  general  review  should 
take  place.  The  multiplication 
table,  of  which  the  elements 
are  known  from  previous  in- 
struction; may  be  written  out 
in   the  following  well-known 
forms,     and     committed     to 
memory  thoroughly. 

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44 


G RUBE'S  METHOD. 


The  pupil  will  understand  from  this  that  the  product  remains 
the  same,  no  matter  in  what  order  the  factors  are  multiplied. 

MEASURING.  —  Oral  Work. 

I.  a.  Counting  up  to  100  and  down  by  2*s,  3%  etc.,  to  lo's, 
beginning  with  i,  2,  or  any  other  number.  The  pupil  must  be 
able  to  do  this  without  hesitation,  and  correctly,  before  this 
part  of  the  course  can  be  considered  finished.  The  written 
form  for  exercises  of  the  same  kind  is  : 


1  +  2  =  3 

3  +  2  =  5 

5  +  2  =  7 

etc. 


4  +  5=9 

9  +  5  =  I4 

14  +  5  =  19 

etc. 


or,  i,  3,  5,  7>  9>  etc- 
4,  9,  14,  19,  etc. 


WRITTEN  AND  ORAL  WORK. 


10  X  10  = 

IOO. 

IOO  =  IOO  X 

=  50  x 
=  33  X 
=  25  x 

=   20  X 

=  16  X 

i 

2- 

3  (+  i) 

4 
5 
6  (+  4) 

i)  IOO 

2)  IOO 

3)  I0° 
4)  100 

5)  I0° 
6)  100 

=  IOO 

=  50* 
=  33  (0 
=  25 
=  20 
=  16  (4) 

*  To  be  read:  a.  2  is  contained  in  100  fifty  times,     b.  Half  of  100  is  50,  etc. 


TEACHING   ARITHMETIC  EXPLAINED.  45 

TOO  =  14  X  7  (-J-  2)  7)  100  =  14  (2) 
=  12  X  8  (+  4)  8)  100  =  12  (4) 
=  ii  X  9  (-f  i)  9)  ioo  =  ii  (i) 

=  10  X  10  10)  IOO  =   10 

=  99  +  1 

=  98  +  2   etc. 

I.  b.  Miscellaneous  exercises  in  addition,  subtraction,  multi- 
plication, and  division,  with  all  the  numbers  below  ioo,  will  be 
a  test  whether  the  pupil  has  the  necessary  mechanical  skill  to 
proceed  to  the  study  of  numbers  above  ioo. 

Examples  like  the  following  should  offer  no  difficulty. 

ADDITION:  14  +  13  +  12  +  11 

X5  +  T7  +  19  +  l8 

25  +  37  +  39  +  J7 

• 

SUBTRACTION  :          90—16  —  12  —  11 
98  -  32  -  41  -  24 

MULTIPLICATION:     3  x  30,  4  x  22,  2  x  44,  2  x  27. 
3  X  25>  35  X  2>  etc- 

DIVISION:                 3)  60,  3)  69. 

4)  60,  4)  72. 

12)  84,  13)  65. 

4)  53- 

A  good  exercise  in  the  combination  of  numbers  is  to  write  a 
series  of  figures  on  the  board,  and  to  direct  the  pupil  to  add  or 
multiply  the  first  two  pointed  at,  to  subtract  the  next,  to  divide 
by  the  third,  etc.  Examples  like  these  should  present  no  diffi- 
culty : 

(3  X  29)  -  (4  X   16)  +  7;   10  X  9  X  3. 

The  teacher  should  always  solve  the  examples  mentally  with 
the  class. 

Grube  recommends  also  the  following  exercises  at  this  part 


46  G RUBE'S  METHOD. 

of  the  course  as  a  test  whether  the  pupil  has  a  clear  or  fixed 
idea  of  each  number  : 

Let  the  pupils  count  from  i  to  100,  but  instead  of  naming 
the  numbers  themselves,  name  two  factors  of  which  each  may 
be  composed.  Hence,  instead  of  counting  6,  7,  8,  9,  10,  the 
pupils  are  to  say  2  X  3,  7  X  i,  2  x  4,  3  X  3,  2  x  5,  etc. 
i  is  to  be  given  as  a  factor  only  in  case  of  prime  numbers. 

Numbers  like  52,  68,  95,  etc.,  must  be  remembered  as  the 
product  of  4  x  13,4  X  17,5  X  19,  etc. ;  and  for  this  purpose 
the  multiplication  table  up  to  20  should  be  studied. 

I.  c.  Somebody  had  $100  and  spent  the  fourth  part  of  it; 
of  the  remainder  he  spent  the  third  part.  What  amount  did 
he  keep?  What  part  of  the  $100  did  he  keep? 

I  have  taken  a  number  3  times,  and  have  4  more  than  half  a 
hundred.  What  number  did  I  take  three  times  ?  • 

Five  times  what  number  is  5  less  than  100? 

Seventy-five  is  three  times  the  fourth  part  of  what  number? 

Exercises  in  changing  compound  numbers  within  the  limits 
of  100  to  lower  or  higher  denominations. 

One  quarter  of  a  dollar  has  how  many  cents  ?  Three  quar- 
ters? 

Half  a  dollar  is  how  many  quarters  ?     Dimes  ? 

A  dollar  is  how  many  dimes  ? 

How  many  months  in  100  days?    Weeks? 

A  hundred  months  are  how  many  years  ? 

One  hundred  pieces  are  how  many  dozen?     Pairs? 

One  year  and  eight  months  are  how  many  months  ? 

One  hundred  ounces  are  how  many  pounds  ? 

Eight  pounds  three  ounces  are  how  many  ounces  ? 

Twenty- three  gallons  are  how  many  quarts  ? 

One  hundred  quarts  are  how  many  gallons  ? 

A  farmer  sold  three  mules  for  99  dollars ;  how  much  apiece 
did  he  get  for  them  ?  etc. 


TEACHING  ARITHMETIC  EXPLAINED.  47 


NUMBERS  ABOVE  ONE   HUNDRED. 


IN  teaching  the  numbers  from  looto  1,000,  the  tran- 
sition is  made  to  the  ordinary  four  processes.  Instruc 
tion  gradually  loses  the  character  of  an  object  lesson,  and 
appeals  to  memory,  understanding,  and  reason  directly. 
Not  that  the  help  of  illustrations  is  discarded  alto- 
gether, for  they  should  be  used  wherever  feasible  ;  but 
when  dealing  with  larger  numbers,  the  only  way  to  il- 
lustrate is  to  show  the  analogy,  with  a  corresponding 
example  in  smaller  numbers,  by  which  perception  is 
enabled  to  help  the  higher  powers  of  the  mind.  A  few 
generalizations  will  be  of  assistance  in  following  Grube's 
idea. 

The  number  100  is  the  last  one  treated  by  itself. 
With  it,  instruction  proceeds  no  longer  from  one  num- 
ber to  the  next  higher  one,  considering  each  number 
separately,  but  deals  with  the  numbers  from  100  to 
1,000  in  general. 

Grube  places  the  work  with  numbers  from  100  to 
1,000  in  the  first  half  of  the  third  year  of  the  course. 
The  first  quarter  is  devoted  almost  exclusively  to  pure 
number,  the  second  more  to  applied  number. 

As  the  relation  of  the  units  and  tens  to  each  other 
has  been  considered  in  the  previous  course,  the  princi- 
pal part  of  the  work  at  this  stage  is  the  measuring  of 
hundreds  by  hundreds,  and  of  hundreds  by  tens. 


48  G RUBE'S  METHOD. 

The  greater  part  of  instruction  here  is  oral  work,  or 
intellectual  arithmetic  ;  written  work  is  but  a  repetition 
of  the  oral. 

In  the  introduction  to  this  division  of  his  work,  our 
author  says,  "As  the  future  study  of  arithmetic  is 
simply  an  application  of  the  insight  gained  by  percep- 
tion into  the  nature  of  the  numbers  from  i  to  100,  the 
following  part  of  the  course  has  for  its  purpose  to  re- 
duce the  relations  of  the  numbers  from  100-1,000  to 
those  of  the  numbers  below  one  hundred,  or,  in  other 
words,  to  show  that  the  relations  of  larger  numbers 
among  themselves  are  of  the  same  nature  as  the  rela- 
tions of  their  elements." 

By  this  practice  the  pupil  arrives  at  the  secret  of  ex- 
cellence in  performing  examples  mentally,  —  the  dealing 
with  numbers  reduced  to  their  smallest  possible  form. 

In  order  to  arrive  at  a  true  idea  of  number,  we  must 
look  upon  number  itself  at  this  stage,  and  not  yet  con- 
sider the  four  processes  as  such.  The  latter  are  re- 
served for  the  second  half  of  the  year.  Intellectual 
and  written  arithmetic  should  always  be  combined. 

As  there  is  no  longer  any  need  for  the  isolated  con- 
sideration of  each  number,  as  in  the  former  part  of  the 
course,  the  only  division  of  the  subject-matter  neces- 
sary is 

A.  THE  PURE  NUMBER  (measuring,  comparing,  com- 
bining}. 

B.  APPLIED  NUMBERS. 

Grube's  six  divisions  of  the  work  with  pure  number 
from  100  to  1,000  show  the  plan  which  he  recommends  ; 
and  after  'having  given  them,  nothing  of  the  peculiar 
features  of  his  method  remains  except  the  teaching  of 
fractions. 


TEACHING  ARITHMETIC  EXPLAINED.  49 


FIRST   STEP. 

NUMBERS     FROM     ONE     HUNDRED     TO     ONE 
THOUSAND. 

Measuring  by  the  units  of  the  Decimal  system,  by  units,  tens, 
and  hundreds. 

Illustrations  should  be  used.  Grube  recommends  solid 
blocks  divided  by  lines  into  10  and  100  units.  Squares  of 
paste-board  will  answer  the  same  purpose. 

EXERCISES  :  768  =  7  hundreds,  6  tens,  8  units.  The  8  units 
belong  to  the  7th  ten  of  the  8th  hundred ;  two  units  would 
complete  the  7th  ten,  3  tens  more  the  8th  hundred,  2  hundreds 
more  would  complete  i  ,000. 

Analyze  in  this  way  500,  704,  174,  714,  829,  999,  etc. 

What  number  has  3  hundreds,  6  tens,  5  units  ? 

How  many  units  in  7  hundreds,  8  tens,  9  units  ? 

How  many  units  in  1,000?  how  many  hundreds? 

Written  Work. 

Hundreds  Tens    Units 

615  =  6  x  100  +  ix  10 +  5x1  =  6         i         5 
204  =  2  x   100  +  o  x  10  +  4x1  =  2         o         4 
or  615  =  600  +  10  +  5  etc. 


SECOND   STEP. 
HUNDREDS     MEASURED     BY    HUNDREDS. 

A.   200  (200;   100) 

(Objective  Illustration,  —  Measuring  and  Comparing,  — 
Rapid  Solution  of  Problems,  —  Combinations  :  The  same  as  in 
the  first  part  of  the  course.) 


5O  G 'RUBE'S  METHOD. 

In  the  first  part  of  the  course  the  diagram  under  the  number 
2  was : 

1  +  1  =  2 

2X1    =    2 

2  —    1    =    1 
2-5-1    =    2 

Hence  the  diagram  of  200  measured  by  ioo  is 

ioo  +  ioo  =  200 

2  x  ioo  =  200 

200  —  ioo  =  ioo 

200   -f-    IOO   =    2 


What  number  is-  contained  twice  in  200?  ioo  is  half  of 
what  number?  What  number  must  I  double  in  order  to  have 
200?  etc. 

B.  a.  300  (300;  ioo)   (300;  200) 

ioo  -{-  ioo  -j-  ioo  =  300 
3  X  ioo      =  300 
300  —  ioo  —  ioo  =  ioo 
300  -T-  ioo      =  3 


(300;  ioo) 


(300;    200) 


20O   +100  =   300 

i  x  200  +  ioo  =  300 

3OO   —    2OO  =    IOO 

300  -f-   200  =    I    (lOO) 


300  is  ioo  more  than  200,  200  more  than  ioo. 
200  is  ioo  less  than  300,  ioo  more  than  ioo. 
ioo  is  200  less  than  300,  ioo  less  than  200. 
300  is  three  times  ioo,  ioo  is  the  third  part  of  300. 

b.  300  —  ioo  —  ioo  -f  200  -+•  ioo  =  ?  etc. 

c.  From  what  number  can  you  take  twice  ioo  and  have 
a  remainder  of  ioo? 


TEACHING  ARITHMETIC  EXPLAINED.  51 

C.  400  (400;    100)   (400;   200)   (400;   300) 

a.  i.  MEASURING  WITH  100.    2.  MEASURING  WITH  200. 


f  100  +  100  -f  100  +  100  =  400 

4  x  100  =  400 

400  —  100  —  100  —  100  =  100 

[  400  -H  100  =  4 


200  -|-  200  =  400 
2  x  200  =  400 

4OO    —    2OO    =200 
400  -r-   200   =    2 


3.  MEASURING  WITH  300.         4.  MISCELLANEOUS  MEASURING. 


!3oo  +  100  =  400 
100  -f-  300  =  400 
i  x  300  H-  100  =  400 
400  —  300  =  100 


400  is  100  more  than  300 

200  more  than  200 

300  more  than  100 

300  is  100  less  than  400 

200  is  200  less  than  400 

100  is  300  less  than  400 


4  is  contained  in  4  once. 

4  is  contained  in  400  a  hundred  times. 

2  is  contained  in  4  twice. 

2  is  contained  in  400  two  hundred  times. 

b.  100  +  200  -f  100  -f-  200  =  ?  etc. 

c.  What  number  is  twice  100  greater  than  200?  etc. 

D.  500,  (500;  100),  (500;  200),  (500;  300),  (500;  400), 


etc. 


E.  600,  etc. 


52  G RUBE'S  METHOD. 


THIRD   STEP. 

MIXED     HUNDREDS     MEASURED     BY     MIXED 
HUNDREDS. 

(This  step  is  a  variation  of  the  preceding  one.  It  is,  of 
course,  neither  possible  nor  necessary  to  consider  every  number 
which  consists  of  hundreds  and  tens,  since  all  that  is  required 
here  is  a  knowledge  of  how  to  perform  the  operation  of  com- 
paring hundreds  and  tens  with  hundreds  and  tens.  For  this 
object  a  limited  number  of  examples  is  sufficient.) 

What  number  is  2,  3,  4,  5  X  no?  440  =  4  X  no, 
=  2  X  220,  660  =  6  X  ?  3  X  ?  880  =  8x?4X?  2X? 
990  =  9  x  ?  3  X  ?  Of  what  factors  may  888  be  considered 
to  consist  ?  999  ? 

If  333  •  •  •  divide  999  among  themselves,  how  much  will 
each  part  be  ?  If  3  divide  999  ?  If  2  divide  888  ? 

Of  what  number  is  120  the  3d  part?  the  4th?  the  5th? 

What  number  equals  the  fourth  part  of  844  ? 

844  is  four  times  what  number?  What  number  is  contained 
4  times  in  844?  Half  of  844  is  how  many  more  than  one- 
fourth  of  this  number? 

One-third  of  333  is  one-sixth  of  what  number? 

Compare  365  with  244.  (365  =  3h  -f  6t  -f-  5u ; 
244  =  2h  -f  4t  -f-  4U ;  3h  —  2h  =  ih;  6t  —  4t  =  2t ; 
5u  —  4U  =  m;  365  —  244  =  ih  -h  2t  -f  lu;  365  is  121 
more  than  244 ;  244  is  121  less  than  365.) 

Difference  between  743  and  1 20  ? 

What  number  is  equal  to  the  sum  of  743  -f-  221  ? 

112  -j-  113  -f-  114  =?     659  —  222  —  124  =? 

in  -f-  212  +  313  =?  etc. 


TEACHING  ARITHMETIC  EXPLAINED.  53 

FOURTH   STEP. 
MEASURING     OF     HUNDREDS     BY     TENS. 

I.  a.  Pure  hundreds. 

If  100  =  10  x  10,  then 

2  x  roo  or  200  =  2  X  10  X  10  =  20  X  10 

3  x  100  or  300  =  3  x  10  x  10  =  30  X  10 

4  X  100  or  400  =  4  X  10  x  10  =  40  X  10 
10  x  100  or  1000  =  10  x  10  x  10  =  100  x  10 

b.  HUNDREDS  AND  TENS. 

If  100  =  10  x  10, 

no  =  (10  x  10)  -f  (i  x  10)  =  n  x  10 

120  =  (lO  X  10)  -f-  (2  X  10)  =  12  X  10 

130  =  (10  x  10)  +  (3  x  10)  =  13  x  10 
990  =  (90  x  10)  +  (9  X  10)  =  99  x  10 

c.  HUNDREDS,  TENS,  AND  UNITS  BY  TENS. 
If  100  =  10  x  10,  then 

101  =  (lO  X  10)  +  I 

109  =  (10  x  10)  -f  9 
906  =  (90  x  10)  -f  6 
814  =  (81  x  10)  +  4 

How  many  tens  in  500,  900?  etc. 

What  number  consists  of  53  tens? 

What  number  contains  9  units  more  than  53  tens? 

How  many  times  10  in  660,  420,  870? 

10  is  the  42d,  66th  part  of  what  number? 


54  G RUBE'S  METHOD. 

II.  Comparison  of  numbers. 

Compare  400  with  900  as  to  the  number  of  tens  they 
contain.  55  tens  are  how  many  tens  less  than  600?  660? 
990  ?  880  is  composed  of  what  4  equal  number  of  tens  ? 
800  -hi  80  -}-  20  =?  210—  160=?  60  tens  are  how 
many  hundreds  ?  What  number  has  8  tens  and  9  units  more 
than  490?  What  number  taken  87  times  and  9  added  to  it  is 
879?  How  many  tens  more  in  73  tens  than  in  twice  240? 
The  looth  part  of  1,000  is  contained  how  many  times  in  500? 
One-third  of  630  is  one-fourth  of  what  number?  The  68th 
part  of  680  -f-  the  24th  part  of  240  are  how  many  less  than 
10  X  36? 

The  exercises  are  followed  by  examples  which  show  that  the 
factors  in  multiplication  are  interchangeable. 

no  =  ii   X   10  =  10  x   ii 

220   =    22    X     10   =    10    X    22 

680  =  68  X   10  =  10  X  68 

What  number  must  I  take  10  times  in  order  to  get  670? 
67  times? 

Of  what  number  is  67  the  tenth  part?  What  is  the  67th 
part  of  670? 

How  many  times  is  79  contained  in  790  ?  What  number 
can  be  taken  ten  times  from  790?  79  times?  79  times  ten  is 
equal  to  10  times  what  number? 


TEACHING  ARITHMETIC  EXPLAINED.  55 

FIFTH   STEP. 
MEASURING    A     NUMBER     BY    ITS     FACTORS. 

I.  a.  Pure  hundreds. 

100  =  2  x  50,  4  x  25,  5  x  20,  etc. 
200  =2x2x50  =  4x50 

200   =2X4X25    =   8X25 

etc.,    etc. 

V 

b.  HUNDREDS  AND  TENS. 

220  =  10  X  22,  and  since  10  =  2  x  5, 

=  2  x  5  X  22  =  2  x  no,  and  since  22  —  2  x  n 
=  10  X  2  x  n  =  10  X  22,  etc. 

f.  HUNDREDS,  TENS,  UNITS. 

426  =  (10  x  42)  -h  6 

=  (4  X  100)  -f-  26,  etc. 

II.  What  is  the  difference  between  980  and  377? 

The  difference  between  980  and  377  is  three  times  what 
number  ? 

By  what  number  must  I  divide  365  to  obtain  five  ? 

What  difference  between  the  22d  and  the  3Oth  part  of  660? 


56  G RUBE'S  METHOD. 


SIXTH   STEP. 

REDUCTION     OF     NUMBERS     FROM    i    TO    1,000    INTO 
THEIR     ELEMENTS. 

It  is  immaterial  in  what  order  the  numbers  are  considered, 
or  what  numbers  are  taken  up ;  the  practice  alone  which  these 
exercises  afford  to  the  pupil  is  important. 

A  pupil  who  has  done  the  work  of  the  previous  course  will 
be  able  to  separate  a  number  into  its  parts  quickly  and  accu- 
rately. The  teacher  gives  the  number,  and  the  pupils  separate 
it  orally  or  in  writing. 

360. 

300  +60  (3  x  100)  +  (3  X  20) 

180  4-  180  3  x  120 

200  -f-  160  10  x  36 

320  +  40  5  x  72 

336  -f  24,  etc.  20  x  1 8,  etc. 


TEACHING  ARITHMETIC  EXPLAINED.  $? 


DIVISION    OF   THE   WORK  ACCORDING 
TO    GRUBE. 


3d  year,  2d  quarter :  Compound  numbers,  money,  weights, 
measures. 

3d  and  4th  quarters,  oral  and  written  work :  Numeration, 
Addition,  Multiplication,  Subtraction,  and  Division  with  any 
number  according  to  the  usual  methods  of  analysis. 

4th  year,  ist  term  :  Object  lessons  in  fractions,  on  the  same 
plan  as  the  lessons  with  the  numbers  from  i  to  10  at  the  be- 
ginning of  the  course. 

2d  term.     The  four  processes  with  fractions. 


FRACTIONS. 

Leaving  the  work  with  whole  numbers,  after  having 
considered  compound  and  applied  numbers  in  the  second 
quarter,  and  passing  over  the  four  species  whose  treat- 
ment is  about  the  same  as  can  be  found  in  any  other 
arithmetic,  we  shall  find  again  an  original  and  peculiar 
application  of  Grube's  idea  in  the  teaching  of  fractions. 

The  pupil  is  expected  to  take  up  this  subject  in  the 
fourth  year  of  the  course  after  having  acquired  some 
knowledge  of  fractions  by  previous  instruction. 

"In  the  same  way/'  says  Grube,  "in  which  the  pupil 
arrived  at  the  perception  of  whole  numbers  by  measur- 


58  G RUBE'S  METHOD. 

ing  them  by  the  smallest  unit,  fractions  are  now  ex- 
plained to  him  by  comparison  with  and  reference  to  the 
number  One,  from  which  they  have  arisen. 

"While  the  number  one  has  appeared  so  far  as  a 
part  of  other  numbers,  it  is  now  considered  as  a  whole, 
which  consists  of  parts.  The  latter  in  relation  to  this 
whole  are  called  fractions." 

As  the  pupils  have  already  learned  to  look  upon 
whole  numbers  as  parts  of  larger  numbers,  the  following 
method  of  teaching  fractions  will  offer  no  special  diffi- 
culty, since  the  process  is  the  same  as  the  one  which 
has  made  them  familiar  with  integers,  and  which  con- 
sists in  the  perception  of  the  manifold  relations  of  the 
number  which  is  being  taught. 

The  order  in  which  fractions  are  considered  is,  halves, 
thirds,  fourths,  fifths,  etc.  The  processes  to  which 
fractions  are  subjected  are  again  : 

I.  Pure  number,  and  under  this 

a.  MEASURING. 

b.  COMPARING. 

c.  COMBINATIONS. 

II.  Application  of  what  has  been  taught  with  pure 
numbers,   in  applied   examples  involving   the    four 
processes. 

The  regular  illustration  for  fractions  is  the  line  di- 
vided into  parts ;  a  circle  divided  into  parts  may  be 
substituted  for  it.  It  is  necessary  to  give  an  abundance 
of  practical  examples  under  each  fraction,  since  the 
four  processes  are  explained  and  made  use  of  at  the 
very  beginning.  In  Division  with  fractions,  Grube 


TEACHING  ARITHMETIC  EXPLAINED.  59 

urges  strongly  not  to  go  here  beyond  the  idea  of 
"  being  contained  in."  It  is  nonsense,  he  says,  to 
speak  of  2  divided  by  one-half,  and  the  like,  at  this 
period  of  instruction.  That  ^  is  contained  4  times  in 
2  will  be  understood  by  the  child,  because  it  can  be 
shown  to  him  ;  but  the  idea  of  division  is  more  difficult. 
Even  examples  like  4  -f-  |  should  not  be  read  four  di- 
vided by  J,  but  rather,  4  is  twice  the  third  part  of  what 
number  ?  or,  still  better,  f  are  contained  in  4  how  many 
times  ? 


FIRST  STEP. 

HALVES. 


I. 


K 


If  I  divide  one  (a  unit)  into  two  equal  parts,  I  obtain  2 
halves.  A  half  is  one  of  the  2  equal  parts  into  which  I  have 
divided  the  whole. 

i  -5-  2  =  \,  or  \  x  i  =  \. 

MEASURING. 

a.  (Addition.)     £  -f  \  =  i. 

b.  (Multiplication.)     i  x  •§•  =  J.     2x^=1. 

c.  (Subtraction.)     i  —  \  =  J. 

d.  (Division.)     £  H-  £  =  i,  i  -$-  }  (£  is  contained  2  times 
in  i). 


60  G RUBE'S  METHOD. 

APPLICATIONS  OF  THESE  FOUR  EXAMPLES  : 

i.  i  -*-  2  =  4»     hence  2  -j-  2  =  |- ,  3  -7-  2  =  J, 

10  -f-  2  =  -ig0,   ioo  -7-  2  =  A-J-Q,  etc. 


tf. 

4  +  4  = 

*i  +  if= 

ii  +  ij  = 

i  +  4  = 

2i  +  i  = 

74  +  44  = 

2  +  4  = 

124  +  i  = 

74  +  8    = 

3+4  = 

184  +  4  = 

74  +  8i  = 

etc. 

etc. 

etc. 

b. 

2      x  |  =  | 

[                             I    X    I 

1=1   x  |  =  i  J 

3X4  =  1 

1   =    2    X    |    =    f   = 

10  x  4  =  J 

#  =  5             3Xi 

4  =  3  x  f  =  1  = 

ioo  X  J  =  Af^  =  50  etc. 

6  X   isi  =  (6  X   15)  +  6  X  i,  etc.) 
9  X  8oJ  = 
(If  i  X  i  =  4,  then  4  X  6  =  f  =  3,  -|  x  9  =  4f,  etc.) 

^.i  —  4=4          2  —  ij=4          2\  —  i    = 

2  -|=  ij  6  -44=  i4  64-3    = 

3  -  \  =  24  9  -  34  =  3^  -  2\  = 

etc.  etc.  8|  —  4!  = 

d.  i  -4-  J  =  2  (for  i  =  f  ,  in  |  one-half  is  contained  twice, 
hence  i  -s-  J  =  2). 

4-^4  =  8  Ii_,_|  =  3          6-5-  ij  = 

6  -5-      =  etc.  -±  -   =  etc. 


2.  ^.  Compare   J  with    i  ;    i  =    i    —    J,     i=i.-f^ 
=  half  of  i,     1  =  2X4- 

<5.  What  number  is  equal  to  the  difference  between  -J  and  i  ? 
How  many  must  I  take  from  16  to  obtain 


TEACHING  ARITHMETIC  EXPLAINED.  6  1 

Of  two  numbers  the  smaller  one  is  9  J,  the  difference  between 
it  and  the  larger  one  is  6J  ;  what  is  the  other  number? 

Name  some  other  two  numbers  that  have  a  difference  equal 
to6J. 

c.  How  many  times  must  I  take  \  in  order  to  have  i  ?  4^ 
in  order  to  have  9?  18?  4^  is  half  of  what  number?  9  is 
twice  what  number? 

The  quotient  is  2,  the  divisor  4}  ;  what  is  the  dividend  ? 
(The  quotient  2  tells  that  4^  must  be  contained  2  times  in  the 
divisor,  hence  the  divisor  must  be  twice  4^  =  9.)  I  must  take 
one-half  of  what  number  in  order  to  have  4  J  ?  etc. 

3.  a.  What  is  meant  by  J  dollar?  dozen?  (One-half  dollar 
is  one  of  the  two  equal  parts  into  which  a  dollar  may  be  di- 
vided.) 

b.  How  many  half  dollars  in  55  cents?    \  dollar  -f-  5  cents, 
etc. 

c.  Difference  between  8  times  55  cents  and  9  times  57  cents? 

(8  x  550  =  8  x  $J  +  8  x  50  =  $4)  40C. 
9  x  57c  =  9  x  $\  r  +  9*X  7C  =  $4%  +  63C 


$5.13  —  $4.40  =  73C,  hence  the  difference,  etc.) 

d.  The  cook  of  a  hotel  buys  17^  pounds  of  meat  +  13^ 
pounds  -f-  8£  pounds.     This  will  be  sufficient  for  how  many 
persons  if  8  ounces  are  the  calculated  allowance  for  each  ? 

e.  If  a  pound  of  tea  costs  ^  dollar,  how  much  can  be  bought 
for  25  cents? 


62  G RUBE'S  METHOD. 

f.  If  5  yards  of  cloth  cost  6  dollars,  what  is  the  price  of  loj 
yards?  (i  yard  =  fifth  part  of  $6  =  $i  +  fifth  part  of  100 
cts.  =  $1.20.  £  yard  =  6oc.  10  yards  =  $12.  10^  yards 
=  $12.60.) 


3.  Applied  examples. 

In  the  treatment  of  the  other  fractions,  the  same 
plan  is  followed.  Fourths,  for  instance,  are  first  com- 
pared  with  the  whole,  then  with  halves,  by  addition, 
multiplication,  subtraction,  division,  and  finally  with 
thirds.  In  the  latter  process,  the  illustration  is  pecul- 
iar, and  consists  of  two  parallel  horizontal  lines  drawn 
close  to  each  other,  the  upper  one  divided  into  four 
parts,  the  lower  one  into  three  parts,  and  then  each 
line  by  light  marks  again  into  twelve  parts,  so  that 
both  show  the  mediating  fraction  of  twelfths  and  their 
relation  to  fourths  and  thirds. 

The  following  is  a  brief  abstract  of  the  treatment  of 
fourths,  giving  in  full  those  details  only  which  cannot 
be  understood  from  what  has  been  said  in  connection 
with  the  treatment  of  J-. 


TEACHING   ARITHMETIC  EXPLAINED.  63 

THIRD   STEP. 
FOURTHS. 

A.  Fourths,  Halves,  and  Units. 

i 1 — -i— 


K 


1.  If  I  divide  i  into  4  equal  parts,  each  part,  etc. 

i  -f-  4  =  J,  or  J  x   i=J. 

«.  |  +  J  =  ?  |  +  J-  =  ?  etc.  (Adding  by  fourths.) 

b.  i   x  J  =  ?  2  x  i  =  ?  etc.  (Multiplying  by  fourths.) 

c.  i  -  J  =  ?  f  —  i  =  ?  etc.  (Subtracting  by  fourths.) 
/£  J  -h  J  —  ?  J  -5-  i  =  ?  etc.  (Dividing  by  fourths.) 

*>.   i  .  Fourths   as   the    quotient    of    integers  :     i  -r-  4  =  J, 
2-5-4,  etc. 

2.  As    the    product    of    fourths    and    integers  :     i    X    3, 
i  X  100  =  ija. 

a.  Addition    (i.    Mixed   numbers   and    fourths,    4!   +   f; 
2.  Mixed  numbers  4-  mixed  numbers,  4|  -|-  4^). 

b.  Multiplication  (integers  X  fourths  and  x  mixed  numbers, 
etc.). 

c.  Subtraction. 

d.  Division. 


64 


G RUBE'S  METHOD. 


B.  Fourths  and  Thirds. 
ILLUSTRATION  : 


,  

|  A 

A 

A 

IA| 

A 

A 

l 

A  j  i1 

a  |  A  |  A 

B 

Or,  if  preferred,  the  circle  rriay  be  used  to  illustrate   the 
same  principle,  as  follows  : 


i.  Fourths  and  thirds  meet  in  twelfths. 


I  =  i  -  A.  4  =  i  +  A- 
1  =  f  x  i  for  j  -  ^  =  f, 
i  =  i  X  |,  for  i  H-  J,  etc. 


=  3  -  4) 


2.  Compare  }  with  f  .     J  =  ^,  |  -  f%. 

i  =  I  "  A,  «  =  i  +  T52- 

i  =  |  x  |  (the  8th  part  of  f  taken  3  times  ;  see  illus- 
tration), for  J  -5-  |  =  |  (the  8th  part  off  (=  g)  is  contained 
3  times  (3-)  ini  =  A  -  A  =  3  -  8). 


TEACHING   ARITHMETIC  EXPLAINED.  65 

f  =  f  X  J,    for  |  -7-  \  =  f .     (The   third  part  of  one- 
fourth  (^2)  is  contained  8  times  in  f .) 

3.  Compare  £  with  f,  etc. 

4.  Compare  halves,  fourths,  and  thirds. 

5..  Fractions,  integers,  and  mixed  numbers. 

6.  \Combinations  and  rapid  solution  of  problems. 
* 

\ 
C.  a    Applied  numbers  -with  fourths. 

b.  Applied  numbers  with  halves,  thirds,  and  fourths. 

c.  Examples  in  analysis. 

d.  Miscellaneous  examples. 

The  other  fractions  are  treated  in  a  similar  way. 

In  giving  an  outline  of  Grube's  method  of  teaching 
the  elements  of  arithmetic,  no  attempt  has  been  made 
to  comment  on  any  part  of  it,  as  it  seemed  desirable  to 
submit  the  whole  system  as  originally  set  forth  to  the 
judgment  of  practical  teachers.  Many  points  are  open 
to  criticism,  and  not  a  few  may  be  obvious  mistakes, 
A  great  number  of  text-books  in  arithmetic  have  been 
written  in  the  country  in  which  Grube's  work  was  first 
published,  which  have  improved  the  original  method, 
and  adapted  it  to  the  special  wants  of  different  school 
systems.  It  seemed  better,  however,  to  present  the 
method  as  it  was  originally  conceived,  without  giving 
expression  to  criticism  and  difference  of  opinion,  and 
to  let  the  well-known  skill  and  ingenuity  of  the  teachers 
of  our  common  schools  adapt  it  to  our  peculiar  wants, 
and  make  such  improvements  and  changes  as  may  seem 
expedient. 

In  regard  to  one  point  of  the  system,  however,  it 
looks  as  if  there  could  be -no  mistake.  The  thorough- 
ness with  which  illustrations  are  used  is  an  indispen- 


66  .  G  RUBE'S  METHOD. 

sable  condition  for  successful  work  in  the  primary 
grades,  If  the  introduction  of  the  kindergarten  has 
taught  some  lessons  to  all  of  us,  the  least  important 
among  them  is  certainly  not  the  remarkable  results  ac- 
complished in  arithmetic,  when  it,is  taught  incidentally, 
by  means  of  the  building-Sfo'ck^S!'  Frcebel's  "gifts." 
The  writer  has  visited  a  kindergarten  in  which  prob- 
lems like  "how  many  twenty-sevenths  in  three-ninths?" 
were  solved  by  children  five  or  six  years  old  without 
any  perceptible  difficulty.  The  explanation  of  this  pro- 
ficiency lies  certainly  in  the  fact  that  ninths  and  twenty- 
sevenths  are,  for  those  children,  not  abstract  terms,  but 
names  of  some  of  the  little  cubes  in  their  toy-box,  and 
that  ninths  and  twenty-sevenths  are  the  names  by  which 
they  know  those  little  objects  with  whose  comparative 
size  long  use  has  made  them  perfectly  familiar.  The 
association  of  arithmetical  ideas  with  perceptible  objects 
alone  makes  arithmetic  intelligible  to  the  child. 

There  can  be  no  doubt  that  many  of  the  methods  of 
instruction  used  in  the  kindergarten  are  excellent  and 
very  suggestive,  and  should  be  carried  over  the  primary 
grades  as  far  as  the  character  of  the  schoolroom,  which 
must  be  kept  distinct  from  that  of  a  kindergarten,  ad- 
mits. In  the  common  school,  children  learn  by  the 
senses  of  hearing  and  seeing  ;  in  the  kindergarten  by 
seeing,  hearing,  and  touch.  The  hand  is  a  very  im- 
portant means  of  education  ;  and  it  seems  evident  that 
pupils  in  the  primary  grades,  who  are  allowed  to  handle 
suitable  objects,  in  arithmetic,  to  count  them,  to  arrange 
them  so  as  to  represent  the  problems  given  to  the 
school,  will  be  able  to  do  better  work  than  if  instruc- 
tion in  this  important  study  is  imparted  without  the 
help  of  objective  illustrations. 


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